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Heres a follow-up game to Nick's Queenless-first-check-wins variant

Graham Waddingham(FM)-Nick Pelling, Correspondence May 2005
1. Nc3 e5
2. Nf3 Bb4
3. e4 f6
4. Bc4 Kd8
5. O-O Nc6
6. d4 Nxd4
7. Nxd4 exd4
8. Bf4 d6
9. Nb5 Bd7
10. Nc7 Kc8 
11. Bd5.
1-0

Its an interesting variant of Chess in that it has remarkably long 
forcing sequences. Black second move 2...Bb4, on the face of it, seems to
be a great move. Knights are *very* strong pieces in the game, and the
pin
seems to win one of whites knights immediately. However, it can never be
taken. The open d-file and activity of white's two bishops would then be
enough to secure the victory. In hindsight Bb4 on move 2 seems to give
white the extra move required to turn the first move initiative into a
forced win. Black is so much on the back foot, that even the smallest
error is enough to finish him off.

Presto Chess: "The first player that gives check with a piece that cannot be taken wins the game."

This leaves it unclear if the checking piece must actually be taken or if it's enough to be able to capture it, and one may instead move the king away or a piece in between. And in the second case: must the capturing move be legal or need it be only pseudolegal?

Another variant of this family:

Like orthodox chess, but a side in check must not move their king. If the check cannot be defended by capturing the checking piece (by a different piece than the king) or moving in between, that side is checkmated and loses. When not in check, one may move the king as usual. Might be called "paralysed king chess".

A milder variant: like above, but the king is allowed to capture a piece attacking him, if it is not a double check.