The professors of my Geometry I course continue to define affine spaces over a vector field V as follows
1) Traslation/Action-like definition Given a $k$-vector space $V$, an affine space over $V$ is a pair $(S, *) $ where $S$ is a set and $*:S\times V\to S$ a function satisfying two axioms
$AS1^\tau$) $\forall P, Q \exists ! {\bf v} : P * {\bf v} = Q$
$AS2^\tau$) $\forall P, Q, R : P * {\bf v} = Q \land Q * {\bf w} =R \to P * ({\bf v} + {\bf w}) = R $
They also proved to us that this definition is equivalent to the one given by a "metric-like" vector-valued binary function on the points of the affine space.
2) Metric-like definition Given a $k$-vector space $V$, an affine space over $V$ is a pair $(S, \delta) $ where $S$ is a set and $\delta:S\times S\to V$ a function satisfying two axioms
$AS1^\delta$) $\forall P \forall {\bf v} \exists! Q: \delta (P, Q) ={\bf v}$
$AS2^\delta$) $\forall P, Q, R : \delta (P, Q)+\delta (Q, R)= \delta (P, R) $
After a first glance at axiom $AS2^\tau$ what came to my mind immediately was that the axiom seems needlessy unnatural as stated and it actually describes an action, an action of the abelian group of vectors $(V, +) $ on the set of points. So I wonder why they never phrased the second axiom as
$AS2^*$) $\forall P \forall {\bf v}, {\bf w} : P*({\bf v} + {\bf w}) =(P*{\bf v}) *{\bf w} $
Looking in my geometry textbook I can read the second axiom presented in the same "form" $AS2^*$) as part of an alternative version of definition 1). In the textbook the main definition given is the metric-like one 2).
Our teachers clearly stated that they would diverge a bit from the textbook by adopting Def 1) instead of the metric-like one.
I suspect that this is part of a teaching strategy so my question here is not what is the reason behind this choice. For this it would be better for me to ask them directly about this choice.
$\mathcal Q1$ My question is rather about the difference between $AS2^\tau$) and $AS2^*$). Is there something that is lost from in the passage from the former to the latter?
In fact, even if I could be able to imagine why they would prefer the "translation-like" over the "metric-like"... $AS2^\tau$) and $AS2^*$) seems to me equivalent and the latter seems to be a more natural way to state it.
$\mathcal Q2$ A broader question can be: what is the proper, formal, way to write those axioms and what are the logical relationship bewen those different but similar versions?
some observations
To me is clear that $AS2^\tau \Rightarrow AS2^*$: by substitution we can recover the identity that says $*$ is an action... but maybe something can go wrong with quantifiers... I'm not sure where and how.
In fact $AS2^\tau $ do not explicitly quantify over vectors because that two vectors are produced by the first axiom in an unique way, but it is clear that they are not free variables even if not bounded by quantifiers. I'm not sure how to write it with proper quantification.
In the other direction it seems we can go smoothly by defining a two new points.