I use "towards lower/higher file" precisely because of this ambiguousness. I will make the definition of "lower/higher file" clearer.
I do think this is a good mnemonic; anybody good at a quick animation
Good idea. An animation would definitely speak a thousand words here.
Finally, I assume when a king can move more than one space, (because of doubled pawns), it still cannot move through check?
Right, it is indeed not quite precise. I'll clarify that since the "long" king moves are sliding moves then moving through an attacked square is indeed forbidden, so that it's consistent with castling.
I use "towards lower/higher file" precisely because of this ambiguousness. I will make the definition of "lower/higher file" clearer.
Good idea. An animation would definitely speak a thousand words here.
Right, it is indeed not quite precise. I'll clarify that since the "long" king moves are sliding moves then moving through an attacked square is indeed forbidden, so that it's consistent with castling.