Please define "sports game" in this context (ie. in the context of TASing, and admission to vault.)

Not to state the obvious, but a "sports game" is a game that depicts the playing of a sport. In the context of TASing/Vault, it is a game that is mostly defined by a timer (i.e, a football game).

Why are "sports games" not eligible for vault? How does this particular game fulfill those reasons?

As I said, most sports games aren't defined by how fast they can be completed. There's a start point and an end point, clearly defined by a timer that needs to run out. This ultimately makes sports games too trivial for Vault, in that there's no actual way to shorten the length of the "game" being played. This is not true for all sports games, of course, this one in particular being an exception, but that doesn't mean the game is complicated enough for site publication.

If the game is pretty linear, and has a definitive and clearly-defined ending, isn't that all that's needed for vault? Why does it matter if the theme happens to be based on some real-life hobby?

It doesn't matter in some cases. We just have a general rule that prevents sports games from being Vaulted since most of them don't exactly apply in the context of speed and obsoletion. What mostly matters here is triviality: Should someone submit a run of an "action" game where literally all you do is hold right and win, we would likely reject that as well for triviality. The fact that this is a sports game means it's just subject to slightly higher criticism in terms of the entertainment value it provides. I'm not necessarily agreeing with the overall rule about sports games, I do actually agree that it should only apply to timer-based sports (football, basketball, soccer, etc) rather than the full spectrum, but as the rule stands now, all sports games have to be given equal treatment. Perhaps we'll discuss this behind the scenes, but as far as I'm aware this is the only time in recent history where this rule has actually been sort of a contested issue.

Would you reject a linear FPS game that just happens to depict paintball rather than an actual war, just because paintball is considered a kind of sport in real life?

A specific paintball game, yes, would be subject to the sports game clause and would need to be entertaining enough to make Moons, but what you're describing is a linear FPS, not a sports game. What matters more for the criteria of "sports game" is how it's presented, not the fact that it contains a sport to begin with. We wouldn't, say, reject a run of Dark Cloud 2 for its golfing minigame, because the primary experience of the game is an RPG. We wouldn't reject your theoretical linear FPS paintball game because, in theory, it wouldn't actually depict the sport itself. It'd just be like turning on the paint cheat in Goldeneye. As it stands, this is just a minigolf game, quite clearly a "sports game" as I don't see any possible way to make minigolf not a sport, and the primary entertainment value is derived from the texture hacking. The only golf TAS that was initially accepted to Vault is, funnily enough, Ninja Golf, which is more of a sidescroller with occasional golf rather than a pure golf game. Krazy Ace was initially Mooned, I suspect it was moved to Vault before we had an official sports game ruling on it, but even if we had the rule back then, the run received good enough votes to make Moons as it was. Like I said, we may revisit and revise this rule in the future, and as such this submission will be re-judged, but it's not exactly a top priority for us. We would need a real borderline case to actually push it up on the list for us, not a joke submission that can't actually be accurately judged off of audience feedback.