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Does anyone know what this "leaps N squares and M over" means? And the Nightknight, do its two N moves have to be in the same direction, or can it change? And can it continue after capture?
Does anyone know what this "leaps N squares and M over" means?
I understand those as 3,1 3,2 and 4,1 leaps. The Nightknight is ambiguous. Without further clarification, I would guess it's NaN.
I don't know, but I read three squares in any direction as a combination of Dromedary and Gecko (or Threeleaper and Tripper, of HG) and the stipulation "and one square over" as an indication that it can jump over one piece (friend or foe) but not over two pieces.
I have no idea what the NightKnight's move is, a conservative interpretation would be a Nightrider restricted to two steps (NN2), but a more free interpretation could indicate a piece resembling a Chu Shogi Lion with Knight moves. Or a Knight version of the Prime Minister (like the Lion, but must stop after a capture).

OK, so it is completely ambiguous. There seems to exist a Zillions file for this variant, though. Could someone who has Zillions test that?
The Interactive Diagram should not have trouble implementing any of the mentioned posibility, but to include an I.D. I would have to know which one to pick.
Edit: Fixed. Thanks AscendingNewAeon!
Nightknight is similar with my piece, Green Mage.
Seems you messed up moves of sen and mid bros

I don't have Zillions, but from what I can make out looking at the source code of the ZRF, Daniel seems to be right: the brothers make camel/zebra/giraffe leaps and the nightKnight two knight leaps in independent directions. My knowledge of Zillions is alas insufficient to be able to tell whether it double‐captures (the code only tests that the intervening space doesn't contain a friendly piece, which suggests it might?) or whether it implicitly forbids null moves (no explicit check seems to be made).
For what it's worth, “two out and one over” seems to be a semi‐common way of describing the knight's move

Another point of doubt: Does the Universal Pawn only capture forward, or is it a Silver General? And does it have some kind of initial double push? (I suppose the Berolinas have that.)

Neither, apparently. Per the ZRF the Universal Pawn moves as a compound of the normal and berolina pawns, moving and capturing one step in any forward direction. All three kinds of pawn can double‐push and they can all capture each other e. p..
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