Perhaps some simple adjectives, "professional bodybuilder", "amateur bodybuilder", "competitive bodybuilder", "recreational bodybuilder" etc could clear up the confusion though. 
probably true
secondly, competitive bodybuilding isnt healthy at all, and there are some of us who have nothing to prove to others whod love to see us travel down their road of misery and health problems. whats the point?
thirdly your baseball batting cages analogy isnt correct, because hitting in a cage isnt "playing baseball"....its hititng in a cage....building your muscles in a gym is
body building, its quite literal.....flexing them onstage ISNT bodybuilding, its just displaying your bodybuilding which happened much prior to that day. so in many other sports you need other players to define the sport (i.e. competition), but in bodybuilding you do not.
you can be a golfer and play by yourself, right? cant you be a skiier and not have to race against others on the mountain? what about a cyclist?
there is no stipulation that requires these activities be competitive so the participant can thus be validated in someone else's eyes.
just as i dont give a shit what someone else is lifting or not lifting in the gym, i could care less who hits their lasix/clen peak on competition day. if beating others on a stage is your bodybuilding motivation, thats cool. if it comes from within, thats great too. but just because you fall in one of these categories doesnt mean the other isnt a bodybuilder.