What precisely is a *classical* spin-1/2 particle?
Talking about Dirac spinors is a distraction; the classical Dirac field has little to do with a single classical particle, just as the classical Klein-Gordan field doesn't have much to do with a single classical spinless particle.
What is the difference between a spinor $\psi_c$ that describes a classical spin 1/2 particle, and a spinor $\psi_q$ which describes a quantum spin 1/2 particle?
Since Baez is talking about quantization, presumably his 'classical' and 'quantum' just refer to classical mechanics and quantum mechanics as usual. That is, a classical system is described by a configuration space manifold and a Lagrangian, or by a symplectic manifold and a Hamiltonian. For a quantum system, the state space is instead a Hilbert space and the Hamiltonian is an operator on that space.
A quantum spin $1/2$ particle has Hilbert space $\mathbb{C}^2$ and lives in the fundamental representation of rotational $SU(2)$. The classical description of a spin $1/2$ particle is not nearly as well known, with some books even claiming that it's downright impossible, because spin is an 'inherently quantum' phenomenon. However, such statements are incorrect; spin is only historically associated with quantum mechanics because both were discovered around the same time. For example, this paper covers the subject primarily in the Hamiltonian formalism, while section 3.3 of Altland and Simons reaches the classical spinor by constructing a path integral for a spin $1/2$ particle and taking the classical limit.
(Soon after this answer was posted, the more insightful answer by knzhou was posted. Please refer to knzhou's answer for clarification about what John Baez meant.)
In the context of quantum theory, the word "classical" is used with at least three related-but-different meanings:
First meaning: A model may be called "classical" if its observables all commute with each other and it is a good approximation to a given quantum model under a specified set of conditions. Example: "Classical electrodynamics."
Second meaning: A model may be called "classical" if its observables all commute with each other, whether or not it is a good approximation to any useful quantum model. Examples: "Classical Yang-Mills theory," and "canonical quantization of a classical model."
Third meaning: A field (or other dynamic variable) may be called "classical" if it is used to construct the observables in a classical model (second meaning). Example: the integration "variables" in the QED generating functional are "classical" fields.
For example, given the action $$ S\sim \int d^4x\ \overline\psi (i\gamma\partial-e\gamma A)\psi, \tag{1} $$ the Euler-Lagrange equation associated with $\psi$ is the Dirac equation. This is a classical model (second meaning) involving a classical field (third meaning). This sounds oxymoronic, because the Dirac equation is often treated as a Schrödinger-like equation in which $\psi$ is the wavefunction, but it can also be treated as the Heisenberg equation of motion for a field operator $\psi$, and this is the sense in which the model defined by (1) is "classical."
Example 1 in the OP shows the generating functional for QED, which has the form $$ \int [d(\text{fields})] \ \exp(iS[\text{fields}] ). $$ The action $S$ in the integrand can be regarded instead as the action of a classical model (second meaning) involving classical fermion fields (third meaning). The model's observables all involve products of an even number of these fields, so the observables are mutually commuting. For this to work, the fermion fields should anti-commute with each other always, not just when they are spacelike separated, just like the observables in a classical model should commute with each other always, not just when they are spacelike separated.
Example 2 in the OP illustrates yet another twist. In this case, the spinors might not be functions of space or time at all; they are not spinor fields (or dynamic variables of any kind). They're just spinors. This is sufficient for introducing things like the relationship between spinors and Clifford algebra and things like the rules for decomposing reducible representations.
By the way, when people talk about classical spinors or classical spinor fields, they might be either commuting or anti-commuting. These are not equivalent, but the word "classical" is used in both cases. This is one of those details that needs to be checked from the context whenever reading about "classical spinors," such as when reading about things like identities involving products of multiple spinor fields.
In this question (which was one of the 1755 questions and answers I got returned after typing "classical spin" in "Search on Physics" and pressing "enter") one can read:
Given a classical spin model, $$H=\mathbf{S}_1\cdot \mathbf{S}_2\tag{1}$$ where $\mathbf{S}_i=(\sin\theta_i \cos\phi_i,\sin\theta_i \sin\phi_i,\cos\theta_i), i=1,2$ is the classical spin.
$H$ is the classical Hamiltonian.
In the Twitter conversation Baez writes:
The phase space of the classical spin-j particle is the sphere with area equal to 4 pi j. The 2-form describing the area element makes this space into a symplectic manifold.
knzhou wrote in his answer:
That is, a classical system is described by a configuration space manifold and a Lagrangian, or by a symplectic manifold and a Hamiltonian.
So because we have a classical Hamiltonian (rather than a Hamiltonian operator) and a symplectic manifold Baez is writing about a purely classical spin, of which he also writes:
For some reason you have to study geometric quantization to learn about the classical spin-j particle, whose quantization gives the more familiar quantum spin-j particle. I don't know why this isn't discussed more widely.
So it turns out (be it in the context of geometric quantization or not) that, contrary to what is taught in many dusty classrooms (the why is very well described by knzhou), the classical spin-$j$ particle does exist and spin is not inherent to quantum mechanics. Baez is very right if he writes that he doesn't understand why this is not known more widely and one has to study geometric quantization to meet these classical spin-$j$ particles.