White is able to capture the Black King, but Black can defend by giving check! Black is obliged by the rules to get out of check first, and cannot end the game by capturing the King.
That is not my (or the Diagram's) interpretation of the rules. The rule is that you must not give the opponent the opportunity to capture your King. When the opponent will never be able to move again, because the game is finished before his turn comes up, he will not be able to capture your King. So capturing his King is just one of the possible ways to get out of check, as it robs the checking piece of the ability to make any move at all..
That is not my (or the Diagram's) interpretation of the rules. The rule is that you must not give the opponent the opportunity to capture your King. When the opponent will never be able to move again, because the game is finished before his turn comes up, he will not be able to capture your King. So capturing his King is just one of the possible ways to get out of check, as it robs the checking piece of the ability to make any move at all..