I'm aware of quite a number of theories of "games" in mathematics. A not-overly-naive preliminary is to ask about what these are, and then what they have in common.

I think the earliest is probably the Polish (Banach-Mazur) type, which have already been defined. "Game theory'' as in von Neumann-Morgenstern is another, important for economists. Combinatorial Game Theory (CGT) is another, with its distinctive ending rule. And then there is the use of "game semantics" in computer science.

The game-like aspect is certainly to do with quantifiers. What mathematicians would recognise as "uniformity" (there exists ... such that for all ...) is to be iterated. My winning is stated in the form "I have a play such that for all plays you make (I have a play such that for all plays you make (...))". This is the typical game quantifier G, and what matters is what happens to the ... . It happens to be true, for example, that any chess game is decided by the time 10,000-ply has been played, so for chess G is a simple iteration, not some sort of "fixed point" construction.

Examples from logic I think largely fall into the "Polish" style of game. That is to judge by the Hodges book on model theory. In CGT there is a no particular distinction between finitary and infinitary games. Whichever theory is being spoken about, you need to explain carefully what the winning or ending condition is, and quite what you mean by a "strategy" (recall that G is sort of non-constructive, being existential in nature).