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I think that color-switching in the sense of true colors is a disadvantage, because it guarantees the piece has no mating potential with and against an orthodox King.

Your first definition seems the correct generalization: if there are no loops of odd length, the accessible squares should break up into two disjunct sets, one set being reachable only after an odd number of moves, the other after an even number. Of course there could be higher-order color switching schemes, like for a piece that only moves N, W and SE. Note that a Ferz is both color-bound and meta-color alternating within the set of accessible squares.

I agree about the FAND.