To me, a sport must have these elements:
1. Significant physical activity (controlling another object that does most of the work, i.e. a car or horse, does not count) and not just a skilled movement (i.e. darts)
2. Competitive
3. Scoring is objective (no "style points"); role of luck in outcome is minimized
The first element rules out spelling bees, chess, poker, etc..
The second element weeds out things like renting a pair of ice skates and casually skating around the rink. Not a sport. (However, competitive speed skating like you see in the Olympics is definitely a sport.)
The third element weeds out activities like gymnastics, ballroom dancing, cheerleading, etc.. While there is clearly a physical component to these activities and they can certainly be competitive, there are subjective and objective elements to the scoring. Therefore, while incredibly difficult at the highest levels, gymnastics cannot be considered a true sport, because there is a subjective component (the judges) in the scoring. Gymnastics has significant elements of sport to it, but also elements of art.
The difference between a "judge" and a "referee" or "umpire" is that the former is usually being asked to make determinations on subjective elements (i.e. artistic points in figure skating), where the referee is usually being asked to make determinations on objective elements (i.e. was the ball out of bounds...the referee might make the wrong call but there is a right and wrong call unlike subjective scoring)
Auto racing: not a sport, fails the significant physical activity test (sorry NASCAR fans)
Poker: not a sport, fails the significant physical activity test and too much luck is involved...I've seen guys on TV play a hand perfectly and then lose because some fluke river card gets flipped. Sorry, that's not a sport.
Figure Skating: not a sport, fails the objective scoring test
Speed Skating: sport (fastest to the finish line wins)
Synchronized Swimming: not a sport
Horse Racing: sport from the horse's point of view, not a sport from the jockey's point of view
Bowling: I say this fails the "significant physical activity" test. Not a sport.
Golf: Close call on the significant physical activity test. I could be convinced either way.
Quarters: not a sport
Surfing: I don't know enough about how it is judged. If it is dependent on the luck of catching a good wave and there are style points involved in scoring, then no.
Ski Jumping: sport
Snowboarding: Doesn't this involve style points? Skill yes, sport no.
Pole Vault: sport (in fact, I'd say most track and field events are about as true sports as you can get, along with weight lifting)
Skateboarding: Are there subjective "style" points in scoring? If so, not a sport.