Axiomatizing orientation in the complex plane
Given your two axioms, the relation is determined for all cycles by its value on cycles of the form $(0,1,z)$, on which it should divide $\mathbb C$ into the upper half plane, the real line, and the lower half plane. For $x \in \mathbb R$ you get the order relation on $\mathbb R$ by considering $(0,1,ix)$. So one way to produce such an axiomatization would be to list all the axioms of the real closed field in this way and then the additional axiom $\mathbb C = \mathbb R + \mathbb R i$. This would no doubt be an ugly axiomatization, but at least it shows that more is possible.
But we can use this as a guide by writing reasonable axioms for the orientation and making sure they imply all the real closed field axioms. For instance if $(z_1,z_2,z_3)$ is oriented and $(z_1,z_4,z_2)$ is not oriented, then $(z_1,z_2,z_3+z_4-z_1)$ is oriented. This implies that the reals are closed under subtraction and hence are a subgroup and that the positive reals are closed under addition.
The ternary relation can be expressed in terms of the triple scalar product: $$ \mathbf x\cdot (\mathbf y\times \mathbf 1)>0 $$ where $\mathbf x=(x_1,x_2,x_3)$, $\mathbf y=(y_1,y_2,y_3)$, and $\mathbf 1=(1,1,1)$.
This suggests some axioms such as those listed under "Properties" in the Wikipedia article.
A too long waste of bits in three parts. The core of the answer is the summary at the end of part 2 section 2.
Part 1. Orientation.
Before treating the aspects on which I agree, let me say one aspect where I disagree (but quite probably only about terminology).
I would not say that orientation in the [complex] plane (your split of non-degenerate triangles into counterclockwise and clockwise is exactly that) is analogous to $>$ on the real field.
It is instead analogous to orientation of the real line (affine one-dimensional real space), where there are two choices and only an arbitrary choice can be made among the two. The same happens in every real affine space of each finite dimension (except 0).
Note that $n$-dimensional affine real space is, from this point of view, not exactly the same as ${\bf{R}}^n$; to identify an affine space with ${\bf{R}}^n$ an affine frame of reference must be chosen, and then an orientation is obtained looking at positivity or negativity of the determinant associated to a simplex. Once a affine frame of reference is chosen, for every other frame of reference one can say "the orientation is the same" or "the orientation is opposite" (positivity or negativity of the determinant of the affine transformation).
On the contrary, positivity or negativity in the real field is not something to be arbitrarily chosen: on a real closed (or even only euclidean) field there is only one compatible order, whose positive cone is the set of squares. "Positive" vs. "negative" for real numbers is not a completely arbitrary choice; instead, it is arbitrary the choice "right" vs. "left" i.e. orientation in real affine (3-)space.
[I have considered affine spaces, but the same happens in vector spaces instead of affine ones]
Part 2. First section: formalization of the structure of "complex plane".
When one speaks of "complex plane", one usually refers to a structure more precise than the field structure on $\bf{C}$. It is the structure or topological field (with the topology of the standard absolute value), or equivalently the structure of field with involution (the standard conjugation). Another equivalent, but commonly not used, choice: the structure of field with a $\sigma$-algebra of measurable subsets (the standard conjugation is the only non-identity field automorphisms which is continuous, or Borel measurable, or Lebesgue measurable, or Baire measurable).
[In the group of field automorphisms there are, assuming choice, infinitely many conjugacy classes of involutions: any real closed field $R$ with continuum trascendence degree over the rationals has a algebraic closure $R(i)$ with the same trascendence degree, hence isomorphic to the complex field; so, conjugacy classes on involutions in the automorphism group of the complex field are in bijection with isomorphism classes of real closed fields of continuum trascendence degree, and one has nonarchimedean such fields whose archimedean part can be an arbitrary real closed subfield of the reals, hence with any trascendence degree from zero to the continuum. So in the complex field one can chose many inequivalent non-standard involutions, or equivalently many inequivalent topologies of nonarchimedean valuations; they are all unsuitable to describe what we intuitively want to be associated to the word "plane". Even the "archimedean" topology is not unique, since by "transporting" the usual one by means of a discontinuous automorphism one has a different, but homeomorphic, topological field structure on the same complex field].
Part 2. Second section: your relation, a little modified.
Once we agree that "complex plane" refers to such a structure (topological field, or field with involution) there is no need to consider an "externally given" orientation: one can define it using the unique $>$ in the self-adjoint real closed subfield.
[This would be possible even for the anti-intuitive alternatives, where the internal and external part of a triangle would both be, from our usual point of view, non-measurable dense subsets of the plane]
So probably your question, when translated in the language above, is almost this: can we use on the complex field (with no other structure) a binary relation "to have the same orientation" among "triangles" (cyclically ordered triples of elements of the complex field) to define the remaining structure to have a complex plane?
As already answered by Will Sawin: this is possible but the axioms would probably be ugly.
Start with a (algebraically closed) field $C$ with a relation "to have the same orientation" between triangles; assume your axioms (1) and (2) or something equivalent to them.
Chose in $C$ one of the two square roots of $-1$, call it $i$. [At the end, one can, and should, check that the other choice would give the same involution i.e. the same $R$]. Consider all triangles $0,1,z$ with the same orientation as $0,1,i$; these $z$ must be, at the end, exactly the numbers of the form $a+bi$ with $a,b$ in the (still to be found!) real closed field, and $b$ strictly positive in it (nonzero square). So the positive elements of the "to be found real closed field" $R$ are the $x$ such that for all $0,1,z$ with the same orientation as $0,1,i$ one has that also $0,1,xz$ has again the same orientation. [Note that it is not sufficient to require that $0,1,xi$ has again the same orientation: consider $x=1-i$].
The first part of the axioms is then: by taking the above defined $x$, together with $0$ and their opposites $-x$, one has a ordered field $R$ where the $x$ are exactly the positive elements, and extending in $C$ this field with the square root of $-1$ gives all of $C$.
This is a finite quantity of first order axioms in the language of fields augmented with the binary relation among cyclically ordered triples of elements of the field.
Once $R$ is recovered, the involution is the only non-identity $R$-automorphism of the quadratic extension $C$ (and conversely the involution defines $R$ as the fixed subfield).
One must require that $C$ is algebraically closed, or equivalently that $R$ is real closed; this is a non-finitizable axiom schema, but if one supposes that the starting structure $C$ was the complex field, this is taken for garanted. Really, this also works more generally for quadratically closed fields $C$ of characteristic 0 or equivalently euclidean fields $R$; in this generality one gains finite axiomatization (but if I remember correctly one loses decidability of the first order theory: I should check Tarski for this).
[If you want exactly the complex field with the usual topology i.e. the usual conjugation, then a non-first order axiom must be added, something like the traditional "archimedean complete", or "locally compact non discrete", or "(locally) connected in the order topology" for $R$ (and requesting real closure becomes redundant). Note that even assuming that the starting $C$ is the complex field $\bf{C}$ one needs to postulate something non first-order about the relation among triangles, or equivalently about the involution, since the "strange" involutions above (and their associated orientations of $C$) have the same first order properties as the standard one]
All in all, a finite number of first order axioms (for the first order part of the theory, and once "algebraically closed" is already given), but I doubt that any simplification can give the same elegance of "algebraically closed field with a fixed involution", which is possibly the simplest axiomatization of the first order part of the theory of the "complex plane".
Summary: I think that the simplicity of this axiomatization (algebraically closed field with involution) naturally discourages the consideration of equivalent alternatives, like the one you almost propose. This might explain why one does not encounter alternate axiomatizations.
Part 2. Transition section to part 3: your relation.
The first order theories of algebraically closed fields of characteristic 0, and that of really closed fields, are both decidable. The two theories are strictly related, but not all of their properties are the same (example: the algebraic closure of a field is unique, but generally not canonically unique; the real closure of an ordered field is canonically, even rigidly, unique; the real closure of a formally real field generally is not unique: there are as many non-isomorphic (as extensions) closures as orderings). The theory of algebraically closed fields with (fixed) involution is "between" the two, but in some way "nearer" to the real theory.
Your proposed theory is almost the same as the theory of algebraically closed fields with involution. Well, the restatement above is definitionally equivalent to that theory, but your "symmetry-breaking axiom" (i.e. the use of a ternary relation instead of a binary relation among triples) introduces a little stronger structure: among the two orientations of the plane one has chosen to call one positive and the other one negative. In terms of the field with involution, one has chosen among the two square roots of $-1$ which one has to be called $i$. I usually call such kind of structure "polarized field" (involution and algebraic closure being already given); I recall having read somewhere this terminology twenty years ago, but I do not remember exactly where (I vaguely remember that it was something somewhat related to "transition amplitude spaces" studied in quantum logic by Gudder and Pulmannova, and related to Feynmann's explanation of the importance of complex numbers in quantum physics as giving a "amplitude" which is more fundamental than the real "probability" since the complex amplitude permits interference. In this context, "polarization" is associated to the physical phenomenon of polarization of waves).
Polarization is what is needed to express unambiguously $C$ as ordered pairs of elements of $R$; in terms of groups of automorphisms and (definitional) equivalence of (first order) structures, one has three levels:
[1] the polarized field $C$ has the same group of automorphisms as $R$ (only the identity, in the standard case), and the polarized $C$ and $R$ are equivalent structures [Corollary: effectively, you propose a equivalent way to look at $R$ by adding structure to $C$, i.e. your relation instead of simply being "analogous" to a relation on $R$ is something that permits complete interdefinability between $C$ with additional structure and $R$];
[2] the $*$-field (field with involution) $C$ has as group of automorphisms the direct product of a two-element group and the group of automorphisms of $R$, and so it is a structure only a little less "precise" than $R$ (one recovers $R$ form $C$, but there are two ways to come back from $R$ to $C$ and no canonical way to chose one way above the other);
[3] the field $C$ has a group of automorphisms usually much larger than $R$, and so is a much less "precise" structure (one still has the two ways to go from $R$ to $C$ but no way to recover $R$ from $C$).
Part 3.
The subtle distinction between the three levels (the complex field, the complex $*$-field, the polarized complex $*$-field i.e. equivalently the real field) has sometimes visible effects.
One example comes from operator algebras and quantum logic. In fact, the $*$-algebras of operators that are used in quantum theory are (almost always) taken to be complex, but the isomorphisms among them that preserve the structure of physical propositions that these algebra define are real $*$-algebra isomorphisms. Or, almost equivalently, complex linear Jordan ring isomorphisms; in usual cases [absence of one or two dimensional nonzero representations, i.e. cases where the associated logic has no classical or "non-contextual hidden variables" components], a Jordan ring isomorphism between the self adjoint parts of the algebras has exactly one extension to a real $*$-algebra isomorphism and exactly one (generally different) extension to a complex-linear Jordan ring isomorphism (which is decomposable in two pieces: a complex-linear $*$-ring isomorphism on one piece, and a complex-linear $*$-ring anti-isomorphism on the other piece). Always in the usual cases above, these isomorphisms correspond (canonically and bijectively with preservation of composition) to the isomorphisms of the associated propositional quantum logics(ortholattice of projections in von Neumann algebras).
So what matters in these cases is that the algebra is "polarizable" (it has a central, anti-hermitian square root of $-1$, which then can play the role of the imaginary unit) but it does not matter what specific polarization is chosen to give a (polarized) complex structure. For "factors" (the center of the algebra reduce to scalars) only two polarizations are possible; in general, there are as many polarizations of the $*$-algebra as central projections i.e. clopen sets in the (compact totally disconnected) spectrum of the center of the algebra. These polarizations i.e. decompositions of the spectrum of the center in an ordered pair of clopens are exactly the above decompositions that split a complex-linear Jordan isomorphism into a associative $*$-ring isomorphism and anti-isomorphism. Changing polarization does not change the physical propositions and their structure. Hence the physical content is encoded by a structure that is a real $*$-algebra (even the $*$-ring alone is enough), the existence of the imaginary unit (i.e. polarizability) being a nice surplus that permits useful constructions, but it does not matter who really is $i$ (provided that a $i$ is available).
Appendix to part 3: basic definitions for $*$-operator algebras.
Since real operator algebras are not so well known (only Li Bing Ren has written books specifically about them) a parallel with the traditional complex case might be useful.
A von Neumann algebra $A$ in $B(H)$ (the $*$-ring of bounded linear operators on a Hilbert space, with the usual adjoint as involution) is the commutant of a self adjoint subset of $B(H)$ ($A$ is the set of all bounded linear operators that commute with all bounded linear operators $b$ in a given set $B$, and also with all their adjoints $b^*$). Equivalent definition: a $*$-subring of $B(H)$ containing all scalar operators and closed in one (hence all) of the usual operator topologies of $B(H)$ weaker than the norm topology.
Closure in the norm topology defines unital $C^*$-algebras, whose abstract characterization is unital Banach $*$-algebras such that $\|xx^*\|=\|x\| \|x^*\|$ (and each $1+xx^*$ is invertible, a redundant condition in the complex case but not in the real case). One abstract characterization of von Neumann algebras is: $C^*$-algebras which as Banach spaces are dual of another Banach space (such a Banach pre-dual is then canonically unique); another characterization: $C^*$-algebras that are Baer rings (the annihilator of a subset is generated by a idempotent, which then can be taken also self-adjoint, and unique such; the ortho-poset of projections is identified with that of annihilators, and so is a complete ortholattice) and have a separating family of positive linear functionals that give completely additive probability measures on the orthoposet of projections (then the completely additive positive linear functionals define inside the dual the positive cone of the unique pre-dual).
What is the relation between real and complex von Neumann, resp unital $C^*$, algebras?
Every complex vN/$C^*$ $*$-algebra is real, forgetting imaginary scalars. [However, not every real vN/$C^*$ algebra comes in this way from a complex one, see below].
A real $*$-algebra $A$ is vN/$C^*$ iff its complexification $A\otimes_{\bf{R}} {\bf{C}}$ is. (The complexification is usually written $A+iA$, but this notation is abusive unless a polarization in $\bf{C}$ was chosen; note that in all this $\bf{C}$ is treated as $*$-field). [However, not every complex vN/$C^*$-algebra comes from a real one in this way, see below].
More precise relations:
A real vN/$C^*$ algebra is equivalent to a complex vN/$C^*$ algebra with fixed involution $J$ ($J$ is not the adjoint operation $*$, but another ring involution which commutes with $*$ and is complex linear; note that $*$ is instead conjugate-linear). Precisely: from the real $A$ go to its complexification with $J(a+ib)=a-ib$; from the complex $*$-algebra with involution $J$ recover the real $*$-algebra as the $J$-fixed elements.
Since not every complex vN algebra admits such a $J$ (Connes, ...) one sees that real algebras are strictly more specialized than complex ones from this point of view. Real $*$-algebras = complex $*$-algebras with a fixed complex involution $J$.
A real vN/$C^*$ algebra is completely determined by its $*$-ring structure (multiplication with real scalars is uniquely definable: algebraically clear for rational scalars, follows by order continuity for real scalars since the elements $xx^*$ form a positive cone and the embedding of the rationals in the self-adjoint part of the center of the unital $*$-ring has a unique extension to a ordered $*$-ring embedding of the reals in the self adjoint part of the center of the $*$-ring).
For complex vN/$C^*$-algebras, the $*$-ring structure is not sufficient: multiplication of $1$ in the $*$-ring by the complex number $i$ gives a central, anti-hermitian square root of $-1$ in the $*$-ring, but conversely any such a element of a vN/$C^*$ algebra induces a (polarized!) complex structure.
So: a vN/$C^*$ algebra over the polarized complex field is the same as a polarized real vN/$C^*$ algebra (polarized meaning that a central anti-hermitian square root of $-1$ has been fixed). Polarizable real $*$-algebra means real $*$-algebra which admits a complex structure, and a complex structure is something a little weaker than a polarization since it only induces a unordered pair of polarizations, without way of choosing one unless the complex $*$-field is polarized.
[Final note: I know that what I have written is far from deep, and possibly also far from intelligent. But twenty years ago I had to waste some time in the disambiguation of such subtle distinctions between "evident" structures, and I like the illusion that what I have written might avoid to someone else such a waste of time]