The tagging situation with fitness centres is a mess. There is a proposal for amenity=fitness_center, but in the talk page people have suggested using leisure=* or even sport=*. In addition, the proposal uses US spelling (center), while OSM tries to consistently use British spelling for tag names (centre).
However, every tag that has been suggested seems to be in actual use in the map, plus some combinations of them.
To illustrate how much of an inconsistent mess we have, here are the taginfo statistics for the different tags: