An interactive experience intended for entertainment.
thosere some long-ass words goddamn
At its core I agree with this.
All games are purely an interactive experience for entertainment, whether it's a ball game, playground game, puzzle, mind game, video game or anything else.
The only defining feature is it being interactive (which can include playing the game or even betting on it, like dog races) and intended to entertain.
Even video games made for Art are designed to entertain the mind.
I think after that a game can be anything. You can tell a story, challenge a player, dazzle with beauty, exercise an individual, earn or lose a player money, or get someone blind drunk.
So long as you have that interactive nature and a design for entertainment then it can go anywhere. The question is a bit like "What do you think a novel is?". As long as it's a story in a book (ignoring precise definitions on length) then it's a novel, and after that it can be about or include anything.