Your definitions are simultaneously pedantic in tone and broadly wrong.
Not all sports are games. Many people would describe (for example) fishing as a sport, but few would call it a game. There's a broad intersection of these categories, but sports are not a subset of games in either a prescriptive or descriptive sense.
Not all games are hobbies. A gladiator may compete in games, but... uh... gladiating was certainly not a hobby. "Hobby" has all sorts of connotations that are not satisfied by many instances of games or gaming. You could call them subsets of "activities" maybe, but certainly not hobby.
I think it's debatable whether competitive gaming is properly called a "sport". From a prescriptive standpoint, some dictionaries give definitions of sport that would be met, other definitions would not be met. Looking at etymologically, you'd assume things would count as "sports" that prescriptive definitions would not consider.
Popularly - descriptively - the term "e-sport" certainly seems to be catching on.
And it's doing so because it's useful in many ways. The substance of these competitions and their supporting organization has a lot in common with "normal" professional sports. There's teams and jerseys and player positions and sponsors. The point of words is to communicate, and calling these competitions sports is communicating a lot of information efficiently, while varying only in one (possibly key) bit - they're not terribly physical competitions (even if they do require a surprising amount of stamina and physical preparation).
Further complicating the matter is that the activities simulated in the game are also often sport-like (though often involving less killing).