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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.