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I can't tell about the specifics of Ice Age Chess and I agree that the fair way to tell would be to ask the inventor, but IMHO DarwinKoala has the right line of thought. A mate is not a mate if you can't actually take the king at the next move.
For an extreme example, consider a hypothetical variant where the queen is not allowed to move a single square (to make a king's move). You play 1.e4, 2.Bc4, 3.Qf3, 4.Qxf7. It would sound odd to claim that it is a mate because it would be a mate in orthochess, wouldn't it ?
As for the unwanted possibility of a king "moving into check", I think it is playing with words. If a king can't be taken at the next move, then its "moving into check" is just a potent optical illusion.
It is always possible to decide otherwise, but I think that in general variant design, playing up to the capture of the king should be the default.
nabla: I'm not sure your definition of checkmate is correct. Fide's website says:
The objective of each player is to place the opponent’s king ‘under attack’ in such a way that the opponent has no legal move. The player who achieves this goal is said to have ‘checkmated’ the opponent’s king and to have won the game. Leaving one’s own king under attack, exposing one’s own king to attack and also ’capturing’ the opponent’s king are not allowed. The opponent whose king has been checkmated has lost the game.
The question in ice age chess would be does the definition of legal move apply at the time of black's 40th or white's 41st? Clearly not an issue in standard chess as the board is the same. I would vote for on black's 40th for symmetry.