Moving king into check Can a player move their king into check if the opponents piece that is able to take king, is putting opponents king in check?
It is never, ever legal to move into check. Period. No ifs, no buts. No exceptions whatsoever.

That's the best way to think about it, since it gives the least scope for confusion. If you want to think of it in terms of capturing the king ending the game, look at it this way: if you move into check, your opponent can capture your king, which immediately ends the game. It's irrelevant if your opponent's capture caused him to move into check too: you can't retaliate by capturing his king because the game already ended.