Sport: Anything two or more people can do together to show that one is better at it than the others.
Marriage is a sport.
But if target shooting can be in the Olympics as a sport, I'm not sure why Pool is any less. For that matter golf seems pretty similar, just on a bigger green.
Heck, I've heard the definition "if you can't play defense it is not a sport" but in pool you actually CAN play defense.
Now if I can just convince my wife learning to shoot pool makes me an athlete....(but she plays the "spouse" sport better than I, so it's hopeless).
By your first definition (which I think you may have been joking), checkers, chess, street fighter II, etc are all sports. I'm with you about shooting. I just refuse to believe pool is a sport. Same for poker, darts, shooting, etc. I get it some people disagree, but that's my opinion. I am always open to hearing why people consider it a sport though!