If it has competition between two or more teams or individuals,and the winner is determined by that head to head competition, then it is a sport. Examples: Water polo, volleyball, tennis, softball, baseball, basketball, auto racing, football, lacrosse, hockey, soccer (OK it's boring), rugby, Aussie football, polo, golf (fun to play, boring to watch),billiards, handball, squash, jai-alai, field hockey (again, boring), motocross, speed skating, etc. If it has competition between two or more teams or individuals, and the outcome is determined by a judge's score, then it is not a sport, it is a pastime. Examples: gymnastics, figure skating, diving, ice dancing, ballroom dancing, synchronized swimming or diving, horse jumping (where the horse jumps obstacles), bodybuilding, etc.
You need to remove horse jumping from that list -- the score is determined on solely objective criteria -- number of obstacles hit, number of refusals and/or falls, and amount of time over the allotted amount. I think you really meant the "dressage", which is subjective scoring.
I also disagree with the main premise that anything with subjective scoring is not a sport -- most team sports would then fall into that categories since referee/umpire decisions are subjective.
It's a sport if it requires athletic ability and conditioning, it's a game if it doesn't. Objective vs. subjective scoring shouldn't have anything to do with whether or not it is a sport.