Let me make a general philosophical point about definitions. A mathematical definition is much more than just its bare semantic content. A definition also carries an implicit intent which must be understood from the way it is used in practice.
For example, the mathematical definitions of directed (multi)graph and quiver are equivalent. So why do some people say directed (multi)graph and others say quiver? It is the intent behind the term: people who say directed (multi)graph want to do graph theory of some kind, whereas people who say quiver specifically want to study quiver representations, quiver varieties, etc.
There are also various examples in category theory of constructions which, on the level of category theory, are formally equivalent (e.g. pullback and fiber product) but which are named after special cases which occur in different contexts. Choosing to use one name over the other evokes a particular context and activates certain intuitions related to whatever you're going to use the construction for.
So, back to your question: strictly speaking the word "associative" is (in my experience) only used to describe a property of a map $S \times S \to S$, so saying "$f$ is associative" is formally equivalent to saying "$f$ defines a semigroup." But these two words occur in different contexts. When you want to talk about a given operation being associative, it is usually in the context of potentially several other operations (e.g. addition in the context of multiplication), whereas when you want to talk about a semigroup, you are usually only talking about the one operation. Failing to recognize this difference in intent will make it very difficult for people to understand you in practice even if you are, on a formal semantic level, saying the correct thing.