The (standard) definition of a group.

Question 2: Here are four examples from my bookshelves:

1. Derek Robinson's A Course in the Theory of Groups, 2nd Edition (Springer, GTM 80), defines a group as a semigroup (nonempty set with an associative binary operation) that has a right identity and right inverses (page 1; he proves they also work on the left in 1.1.2, on page 2).

2. Marshall Hall, Jr.'s The Theory of Groups (AMS Chelsea Publishing is the version I am looking at). Gives both the two-sided and the one-sided versions as part of the overall definition of a group, indicating the two-sided one is "redundant" and giving the one-sided one.

3. Burnside, Theory of groups of finite order (I'm looking at Dover's 1959 reprint of the 1911 second edition of the original) defines a group asking only that elements have left inverses. He does not require an identity explicitly, but it follows from the requirement for left inverses.

4. van der Waerden's Algebra requires only a left identity and left inverses.

Question 1. If you want to fit groups into the more general theory of semigroups, monoids, etc., then note that in a monoid you must specify that the identity is two-sided; you cannot deduce it. So if you want to say something like "A group is a monoid in which every element has inverses", or "A group is a semigroup that has an identity and also has inverses for every element" (this is basically what Bourbaki does), then your specification of identity must be two-sided, in which case having the inverse be defined one-sided looks a little strange.

Also, the definitions match the experience that most undergraduates will have had: at this point, they are probably familiar with the usual numerical examples ($\mathbb{Z}$, $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}$, maybe the quaternions, possibly the integers modulo $n$), and possibly also matrices. So it's best to give a definition that matches expectations even if it is a bit more involved/superfluous, than one that is formally more inclusive (by putting fewer conditions on the object) but which may seem to invite queries when matched with the usual examples. Especially when one almost immediately would show that they are in fact two-sided.

I cannot answer your second question, but I'll try answering your first.

Even if we were to work with the second definition, you can bet one of the first things any textbook author would do would be to prove that the second definition implies the first and then work with that throughout the rest of the text.

Also, the fact that the two definitions are equivalent is not really seen as important, because it is rarely the case that we start with a structure knowing it satisfies the requirements of the 'right-group' definition but unsure whether or not it is a 'true group'.

In addition, a definition like this is likely to cause more confusion to the reader. While it may be superficially leaner, people find symmetries very intuitive. Defining groups in this way makes them seem strange (even more than they already are to a lot of people) and raises a lot of questions along the lines of 'what if this rule is tweaked?' which are perfectly fine questions to ask, but they detract from what most abstract algebra books are about.

Ultimately, I think if this change in definition was introduced in a book, it could give the reader some interesting questions to think about, but will mostly just take up some extra space towards the start of the book exploring things which will be irrelevant to the rest of the book, after which the author would go on ahead using the standard definition anyway. Which is probably why it isn't used much.

(feel free to comment or edit for any corrections or suggestions)

Technically speaking, neither of your definitions is correct, because (iii) refers to an undefined $e$. From a rigorous point of view, you have 2 options:

Option 1

$(G,*,e)$ is a group iff $*$ is a binary operation on $G$ and $e∈G$ and:

1. $∀a,b,c\ ( \ (a*b)*c = a*(b*c) \ )$.
2. $∀a\ ( \ a*e = e*a = a \ )$.
3. $∀a\ ∃b\ ( \ a*b = b*a = e \ )$.

Option 2

$(G,*)$ is a group iff $*$ is a binary operation on $G$ and:

1. $∀a,b,c\ ( \ (a*b)*c = a*(b*c) \ )$.
2. $∃e\ ( \ ∀a\ ( \ a*e = e*a = a ∧ ∃b\ ( \ a*b = b*a = e \ ) \ )$.

~ ~ ~

That aside, it is wrong to think that it is at all good to have the 'leanest' definition possible. For example, PL (propositional logic) can be axiomatized by a single axiom sentence schema (e.g. Meredith's as given on wikipedia). If you want to make things worse, use the sheffer stroke (NAND) and no other boolean connectives, just because NAND is functionally-complete.

Another example is PA (First-order Peano Arithmetic). The axiomatization of PA in terms of a discrete ordered semi-ring with induction is far superior to the successor-based axiomatization, simply because it reveals more of the true structure of the intended model $ℕ$ of PA. In fact, the motivation for PA in the first place arose from wanting to axiomatize $ℕ$, so we care only about theories that can prove the basic properties of $ℕ$, and unsurprisingly these basic properties are expressed precisely by the discrete ordered semi-ring axioms plus induction.