[QuesnayJr]

Commutative and abelian aren't really synonyms. "Abelian" is reserved for objects that have a certain amount of rigidity. Commutative monoids are squishy, while abelian groups very rigid. Another place you'll see the name "abelian" is "abelian Lie algebras", which are also rigid. "Abelian categories" axiomatize the kind of rigidity abelian groups have.

Magmas are usually called "groupoids", but there's another generalization of group also called "groupoids". I'm actually not sure they really deserve a short name, rather than just "set with a binary operation", since there isn't much you can say about them in that generality that you can't generalize to "set with two binary operations", "set with a binary and a trinary operation", etc. The argument for a name is it gives you something to modify, since there are interesting special cases such as "medial groupoids". (An example of a medial groupoid is the real numbers with the "average of two numbers" operation.)