How can you deliver checkmate with a king?

Here is another rule, this time concerning what check requires.

12. Check
12A. Definition.
The king is in check when the square it occupies is attacked by one or more of the opponent’s pieces; such pieces are 
said to be checking the king. Check is parried (a player gets out of check) by capturing a sole checking opposing 
piece, interposing one of the player’s own pieces between a sole checking piece and the king (not possible if 
checking piece is a knight), or moving the king. The king cannot parry check by castling (8A4

This is why the bishop is not checking (or checkmating) the opposing king. Ifi it were, the threat could be parried by 1) capturing the bishop, 2) interposing one of his own pieces between the bishop and king, or 3) just moving out of check from the bishop. It should be pretty obvious none of those will work. Why? Because it's not the bishop that's attacking the square the enemy king occupies.

Again you've gotten check and checkmate confused. Although a check is simply an attack on the king (In the above example, the rook is checking), a checkmate is an entire position that every piece contributes to. So although, in the checkmate position, the rook is checking, the bishop is checkmating because the bishop officially changes the position from a "playing" position to a "checkmate" position.

In most checkmates, the checking piece is the same as the checkmating piece, but not in a discovered checkmate!

YES!! I think you are seeing the light. In most checkmates the checking piece is the same as the checkmating piece, but not in a discovered checkmate.

In that case the checkmating piece is revealed. It's a discovered attack, a discovered checkmate. The piece that moves (in this case the bishop) reveals the checkmating piece. At least that is what the rules of chess dictate anyway.

???

First, you said that it is true that in a discovered checkmate, the checking piece is not the same as the checkmating piece. In the diagram below, the rook is the checking piece, while the king is the checkmating piece, because the king's move checkmated black.

Then you contradicted yourself by saying that, in a discovered checkmate, the checkmating piece is the one that is revealed, not the one that's moving, meaning che checkmating piece is the same as the checking piece, which is not correct in the case of a discovered checkmate.

In most checkmates the piece that moves (and checks) is the checkmating piece. I thought that was what you were referring to. But not in discovered checks or checkmates. In a discovered checkmate the revealed piece is the checking piece (and therefore the checkmating piece). Sorry for the confusion. I thought you were recognizing most checkmates do not happen with discovered attacks.

So in your diagram, post 168, the king moves to reveal the checkmating rook. At least according to the rules of chess.

Please link your source. Because my source states that a piece that moves to cause checkmate checkmates.

Another thing: The state of checkmate is contributed to by all pieces on the board. However, the action of checkmating, or, according to the dictionary: "To put [a king] into checkmate", is done by the piece that moves, provided that by moving, that piece puts the enemy king in checkmate.

I already did. USCF rules 4 and 12.

I even copied them, word for word.

I said LINK your source, not quote it.

Well I'm sure you have access to the internet. Just google USCF rules. Or USCF rulebook. It will take you to uschess.org.

This topic comes up every 2 weeks, there´s a search function here on the forums.

Here you go: Morphy against his father:

Yes. And in this final diagram, the rook is checking the king. The white king moved of course, but the white king never checked the black king. It revealed the checkmating piece, much like a discovered check, or possibly a promotion or en passant would.

Nope. If the King hadn´t made the castling move, the R would not have been able to mate. So the K was the active mating piece.

Here we go. Nowhere, in this entire PDF, does it say anything about which piece checkmates. All it says is that a checkmate is when a king is attacked and has no escape, and that the checkmated player loses.

https://new.uschess.org/sites/default/files/media/documents/us-chess-rule-book-online-only-edition-chapters-1-2-10-11-9-1-20.pdf

And in this rulebook that you have provided, 4A says this.....

4. Objective and Scoring 
4A. Checkmate.
The objective of each of the two players in a game of chess is to win the game by checkmating the opponent’s king. 
A player’s king is checkmated when the square it occupies is attacked by one or more of the opponent’s pieces and 
the player has no move that escapes such attack. See also Rule 12, Check; 12C, Responding to check; and 13A,

And because it says that, it DOES specify which piece checkmates. It's right there in black and white. "a players king is checkmated when the square it occupies is attacked by one or more of the opponents pieces". The piece that attacks is the checkmating piece, per the rules. NOT the piece that reveals the discovered attack. How do we know this? Because there is anothe rule, responding to check. All the responses involve the attacking piece, not the piece that gets out of the way.

CHECKMATE IS CHECKMATE!

In your rulebook, you will also notice the conditions for check, and responding to check. They involve 3 things. Capturing the attacking piece, interposing between the king and attacking piece, or simply moving out of check.

So in our examples none of these would get the king out of check. Why? Because the piece that moved is not attacking the king. The responses to check are to the piece that attacks the square the king occupies, NOT the piece that revealed the attack. For example, none of those responses to check would work for the bishop would they?

The rules are rules. And they are clearly understood by "almost" everyone who plays chess.

The rules say there are three ways to escape (or parry) checkmate. Capture the attacking piece, interpose between the attacking piece and king, or move out of check. Which of these three ways to avoid checkmate would work for the castling checkmate in your example?

If the checkmating piece is the king, can you capture the attacking king? I'll say no. Can you interpose between the two kings? I'll say no. Can you move out of check from the attacking king? I'll say no. The reason it's no to all of them is because a king cannot attack (check) another king. It's against the rules.

But, I will give you credit. You are right, the rook is the mating piece.

Fiction. You can't read. Nowhere it says which piece checkmates the king, only that a piece attacks the king (which is obvious because at least one unit gives check). But check is just one aspect of checkmate and does not speak for all of it!

I think what you are referring to is this: "A player’s king is checkmated when the square it occupies is attacked by one or more of the opponent’s pieces and the player has no move that escapes such attack." Please read your quote very closely. This is describing the state of checkmate, not the action. But I see how you got confused.