I don't completely understand what situation / position you talk about. Is this all in the move diagrams, or is it when you are moving around pieces in the original board diagram? So the following commenst might not be to the point.
1) An X means the King is not allowed to go there, for reasons of check, but indicates that it would otherwise have been a legal destination. (So that indeed it could deliver check there, as for capturing a King you are allowed to move into check in orthodox rules.) When you say 'protected by a Pawn' you mean 'attacked by an enemy Pawn'? I don't understand what the other King has to do with it.
2) I suppose this is about the King's move diagram, when 'materializing' the Pawn by hovering over a square. This diagram shows the three forward slide as red diamonds, which is the marker symbol used for 'capture only'. That is indeed to generous, because it should really be 'King capture only'. (In Betza notation: k mode rather than c mode.) I only could imagine so many marker symbols, and I did not have a dedicated marker for this particular mode (which AFAIK only occurs in Xiangqi and Eurasian Chess). So I used the same marker as for normal captures.
When you actually put an enemy Pawn in the path by hovering, you will see that the Pawn cannot be captured. I don't understand what you mean by "extra capturing move". As far as I could determine the move diagram highlights all squares up to, but not including the Pawn with red diamonds. When selecting a King in the main board (rather than viewing it in the move Diagram) only the actual pseudo-legal moves will be indicated. So a square reachable by the forward slide will never get highlighted, unless the enemy King is on that square.
Come to think of it, though, the X would have been a more logical choice to indicate these forward slides, as moves that are forbidden because they stumble into check also effectively do have k mode, as they can still be used to capture a King.
What is really needed here is a better specification of what the symbol means; the standard legend (summoned by clicking on 'here' two times) does not mention that the red diamond has an ambiguous meaning in this case. This could be solved by writing the extra sentence (under the clickable list of pieces next to the board) "The forward distant moves in the King's move diagram can only capture a King."
3) This is a general problem with pieces that have a location dependent move: these would need different move diagrams for each location. The move diagrams always give the 'nominal move' (i.e. unrestricted by confinement), and the board of such a diagram should not be considered the real 'absolute' board, but rather a 'relative board' centered around the piece, wherever that may be. It would be far more confusing to omit the King's three forward moves in that diagram; people would think the King could never move forward, no matter where it is.
I don't completely understand what situation / position you talk about. Is this all in the move diagrams, or is it when you are moving around pieces in the original board diagram? So the following commenst might not be to the point.
1) An X means the King is not allowed to go there, for reasons of check, but indicates that it would otherwise have been a legal destination. (So that indeed it could deliver check there, as for capturing a King you are allowed to move into check in orthodox rules.) When you say 'protected by a Pawn' you mean 'attacked by an enemy Pawn'? I don't understand what the other King has to do with it.
2) I suppose this is about the King's move diagram, when 'materializing' the Pawn by hovering over a square. This diagram shows the three forward slide as red diamonds, which is the marker symbol used for 'capture only'. That is indeed to generous, because it should really be 'King capture only'. (In Betza notation: k mode rather than c mode.) I only could imagine so many marker symbols, and I did not have a dedicated marker for this particular mode (which AFAIK only occurs in Xiangqi and Eurasian Chess). So I used the same marker as for normal captures.
When you actually put an enemy Pawn in the path by hovering, you will see that the Pawn cannot be captured. I don't understand what you mean by "extra capturing move". As far as I could determine the move diagram highlights all squares up to, but not including the Pawn with red diamonds. When selecting a King in the main board (rather than viewing it in the move Diagram) only the actual pseudo-legal moves will be indicated. So a square reachable by the forward slide will never get highlighted, unless the enemy King is on that square.
Come to think of it, though, the X would have been a more logical choice to indicate these forward slides, as moves that are forbidden because they stumble into check also effectively do have k mode, as they can still be used to capture a King.
What is really needed here is a better specification of what the symbol means; the standard legend (summoned by clicking on 'here' two times) does not mention that the red diamond has an ambiguous meaning in this case. This could be solved by writing the extra sentence (under the clickable list of pieces next to the board) "The forward distant moves in the King's move diagram can only capture a King."
3) This is a general problem with pieces that have a location dependent move: these would need different move diagrams for each location. The move diagrams always give the 'nominal move' (i.e. unrestricted by confinement), and the board of such a diagram should not be considered the real 'absolute' board, but rather a 'relative board' centered around the piece, wherever that may be. It would be far more confusing to omit the King's three forward moves in that diagram; people would think the King could never move forward, no matter where it is.