Also despite being named Human chess it's a form of chess where a computer is absolutely necessary.

Should that always be the case, though? We could try to force a position where 2 separate moves checkmate. Then only 1 (presumably the one that results from capturing the highest valued piece?) would be the engine result.

If you can check your opponent, giving them only one legal move, you win (because it is the top engine move).

This is a good point. If you can check with your queen but hang it, the “best move” would be to take it. Make it so they have to take it, for example in a back rank, and you win.

This is explicitly addressed:

  When multiple moves have the top score, they are all top moves,

I wonder if a move that checkmates is scored lower than a move that checkmates and captures.

Forced mates are generally scored with the number of moves to mate, e.g. "M2"

So you can start down a sequence that gives check-mate, but once it is the shortest sequence to check-mate, you have to abandon the check-mate.

So you can never actually play a mate in one.

Checkmate .. with advantage!

With this variant you can win without checkmating: just checking with leaving only 1 forced move is enough to win the game.

As long as putting the king in check isn't optimal…

Which it quite often is. Like the example they show with the early-game check with the queen, putting the king in check by placing a totally undefended piece within one square of the king is (usually) a suboptimal move, and the king taking that piece will (often) be the most optimal way of getting out of check.

In this variant the attacking piece will effectively be protected by how bad its move was. Creates some interesting incentives - the only way to checkmate is a move that is normally not optimal and has only one way out of check.

Right, so your check should be in a context where another move was mate in 1

Not necessarily. The check with forced answer could be a terrible move in normal chess. E.g. a check with the queen where the queen can simply be taken by the king.

Aren't there situations where 2+ moves cause checkmate? Only one can be the top engine move. Or are all of those effectively impossible to reach unless your opponent helps?

It says at the bottom:

> When multiple moves have the top score, they are all top moves, even if visual markers (like move arrows) suggest the engine prefers one over the other.

Since all moves that checkmate the opponent will have the same score (M1 or -M1) they'll all be illegal.