I would not encourage you to introduce a new terminology, for two reasons.
First, it would increase the confusion between existing terminologies (see below). Secondly, it could make it difficult to find relevant information.
There is a large litterature on Semigroups. The free semigroup on a set $A$ is denoted by $A^+$.
Idempotent semigroups have been studied for a long time and bands is another well established terminology for them. In particular, it is known that every finitely generated free idempotent semigroup is finite (a nontrivial fact, as emphasized by Andreas Blass' example, see [3] for an efficient algorithm). Moreover, a complete classification of the varieties of idempotent semigroups is available [1].
Commutative semigroups are also well studied, [2] is an excellent reference. Idempotent and commutative semigroups are also known as semilattices. The free commutative semigroup on a set $X$ is denoted by $F_X$ in [2], but this is a context-depending notation: $F_X$ or $F(X)$ could be used for the free object on $X$ for any algebra.
Magmas are sometimes called groupoids. See your own question for a notation of the corresponding free algebra. Idempotent magma is a very natural name: it is used for instance in two answers to this question. Commutative magmas have their own wikipedia entry (rock, paper, scissors being the emblematic example). Commutative and idempotent magmas are used in this thesis.
[1] J. A. Gerhard, (1970), The lattice of equational classes of idempotent semigroups", Journal of Algebra, 15 (2): 195–224
[2] P. A. Grillet, (2001), Commutative Semigroups, Springer Verlag, ISBN 978-0-7923-7067-3
[3] J. Radoszewski, W. Rytter, Efficient Testing of Equivalence of Words in a Free Idempotent Semigroup. SOFSEM 2010: Theory and Practice of Computer Science. SOFSEM LNCS 5901, Springer (2010) 663-671.
Best Answer
Yes.
According to Theorem $1$ of
Minimal Identities for Boolean Groups
N. S. MENDELSOHN AND R. PADMANABHAN
JOURNAL OF ALGEBRA 34, 451-457 (1975)
a structure $\langle A, \star\rangle$ satisfies the single identity $(x\star y)\star (x\star (z\star y)) \approx z$ if and only if the structure is an abelian group of exponent $2$.
[The fact that such an identity exists follows from work of Higman and Neumann from 1952, but the paper I cite gives short identities.]