First, let me mention that the ambiguity I wondered about is now resolved.

Turning to Larry's puzzlement over the logic behind Parton's rules, I
think the logic lies in what makes for best gameplay. There are two
extremes that each seem more consistent than Parton's choice. One extreme
is to count check only on the completion of a move, and the other is to
never allow any move that leaves a King in check before the transfer of a
piece to the other board. In contrast to these two internally-consistent
options, Parton chose to count check only when a move puts a King in check
before the transfer is made, and to not count check when the King is
already in check and the pre-transfer move does not eliminate the check. I
think Parton made the right choice, and here's why I agree with it. The
first option I described, of counting check only when a move is completed,
would make it too easy for a King to escape check. The second option, of
always counting check before the transfer, would make it too difficult to
escape check. In fact, it would remove all possibility of blocking a
check. Any move that blocked a check before the transer was complete would
fail to actually block the check, because it would be transferred to the
other board, where it no longer blocked the check. To make it neither too
hard nor too easy to escape check, the right choice is to not allow any
move that puts one's own previously unchecked King into check, while
allowing moves that merely postpone the elimination of a pre-existing
check until the end of the full move.