I consider neither fishing nor golf to be a sport. You can make an argument that athletic skill (such as hand-eye coordination and sometimes raw strength) plays a sufficient role in fishing, and it's clear that precision and strength both play a significant role in golf. However, by my definition these are both athletic competitions, not sports. Darts is also an athletic competition, not a sport. Same for diving, ballroom dancing, and any stay-in-your-lane racing.

Billiards I would consider a sport. There's clearly an athletic component which plays a major factor in the outcome; strength, accuracy, and precision are all important.

In addition, there's a concept which I believe is crucial to separating sports from other competitions: defense. In any sport, you must be able to do something within the rules to defend against your opponent's ability to win. In baseball, players field the balls that are hit. In football, defenders tackle offensive players. Even in some races, you can position your body such that other competitors have to expend additional energy to pass around you. By contrast, the other athletic competitions mentioned above have no notion of defense. A 100m sprint could have one participant in New York and the other in Tokyo, and assuming that all event conditions could be accurately replicated the race would be completely fair. Same goes for golf; you can't block an opponent's shot with your own ball or do anything else (within the rules) to disrupt your opponent's ability to complete a hole.

Car racing meets both of the above conditions: must have an athletic component, and must involve a notion of defense.

I'll take care of the incentives for my children myself.