So are "all sports games" but "not all games are sports"?
I don't know, things like track or swimming are clearly sports but don't quite seem like games. Feels like it depends on the complexity of the objectives and how you're allowed to reach them. Traveling in a straight line as fast as possible just doesn't clear the bar for me.
Tough distinction to draw regardless, but basketball clearly qualifies as both.
EDIT: Reading the OP's question more carefully, I'd say what happens on the court is more of a sport, though it's got elements of both, but what happens behind the scenes seems more like a game (it's not a sport, at least). So maybe it depends on what level you're looking.