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Justaminute: You are right, there are basically two ways to solve such questions.
1) Defining what is meant by check and checkmate in every new variant (even in Loop Chess, you will want to make clear that a ortho-checkmate from a distance can by parried by dropping a piece).
2) Going back to the axioms and work out what chess really means by check and checkmate, then keep those axioms as they are in every new variant bar some exceptions (three-check chess would be one).
I am not saying that 1) is stupid. Actually it is the way things are most often done. But even if you want every variant to include a (generally redundant) part about check and checkmate in its rules, it doesn't hurt to know that there is a default value given by 2).