This is a distinction in terminology that has no effect on gameplay.

I don't know. One time I played a blitz game where extreme time pressure left me getting my king chased across the board. I moved it next to my opponent's king and then captured his king with mine on the next move.

Presumably blitz rules didn't actually allow you to move your king into check like that, but maybe blitz games more routinely include illegal moves that are never detected?

Edit: there is a relevant distinction like this according to FIDE!

http://www.fide.com/component/handbook/?id=171&view=article

In normal play, when "it is found that an illegal move has been completed, the position immediately before the irregularity shall be reinstated." (7.5a)

But in rapidplay (including blitz), "[i]f the opponent does not claim and the arbiter does not intervene, the illegal move shall stand and the game shall continue." (A.4)

The FIDE laws still seem to assume the presence of an arbiter, which wouldn't always be the case for blitz games outside of a tournament or other formal event. But I guess there is an official notion even according to FIDE that illegal moves (including moving into check!) can be allowed in a blitz game if nobody notices them, which is not the case in a standard game.

Nonetheless, for all types of games, "[l]eaving one’s own king under attack, exposing one’s own king to attack and also ’capturing’ the opponent’s king are not allowed." (1.2)

Okay, you're right. I didn't think of that kind of situation arising, and that's awesome.

It does, because of stalemate. In normal chess that is a draw, in chess-with-king-capturing it's usually a loss.

And stalemate is huge, e.g. many K+pawn vs K endgames are drawn because of stalemate, without stalemate a one pawn advantage would be much more significant.