Sometimes you pick from a set of actions for each you have equal reasons/information to pick. And you pick technically randomly. But after you made the choice, you stick/commit to it, you identify with it. The other choices are no longer a possibility. You become fixed in this particular causality. And it defines "you" as "the one who [randomly] picked choice X and Y and ... and ended up on this particular causal path".

This is what exercising free will means and how it defines you.

And if you put it in this terms it also becomes obvious that any kind of mechanism can "exercise free will" if you choose to look at it from this perspective.

(The problem with defining "free will" I think is that people get stuck thinking in terms of it as a fixed concept that needs a "what is" type of definition, instead of a procedural/operational "how to" definition. You only care about processes and mutations to the state of reality... not fuzzy wozzy philosopondering. Replace "what is" kind of knowledge with "how to" kind of knowledge and concepts like "free will" become christal clear and obvious. Also it's pointless to bring in another undefined concept, eg. "soul" when defining other.)