in chess lingo the most common is “legal moves”

in chess lingo, most common is "moves"; only in a weird circumstance (beginners?) would you need to say "legal". the "possible" qualifier would probably be used to suggest "future" moves as opposed to the moves already made to get to a position. it would be more likely for whatever reason to say "how many possible moves" than "how many future/hypothetical moves", i.e. the use of possible is not to rule out the idea of impossible, simply to mean how many "could you make now from a particular position"