Chess rules question

I ran across this question recently and was amazed to have to admit that I didn’t really know the rule. Here’s the situation:

White moves (say) his bishop, putting black in check. Black moves his knight onto the bishop’s diagonal, interposing it between the bishop and his king and thus blocking the check. But the white king happens to be positioned such that this knight move appears to put the white king in check.

Yet does it? That black knight is pinned: it can’t move without leaving the black king in check (by the bishop) and thus can’t really be viewed as a valid threat to the white king.

(Bonus points if you can cite the actual chess rule that resolves this.)

Yes. Check by interposing piece, rule 12D.

Just to quote the rule itself

12D. Check by interposing piece.
A piece blocking a check to the king of its own color, commonly referred to as interposing, can itself give check to
the enemy king.

Is the now-checked player obligated to get his king out of check?

Yes, according to Rule 12C (thanks, Maserschmidt):
Check must be parried on the move immediately following.

Of course it can move. If it takes the white king the game is over, it doesn’t matter if the black king is in check.

To state the obvious, the black knight can’t move because it’s white’s turn. What happens on black’s next move, including whether or not they can move that knight, depends on what white does to get out of check.

FIDE (the International Chess Federation) also has a similar rule about a pinned piece giving check:

The rule may be found here: FIDE Handbook E. Miscellaneous / 01. Laws of Chess / FIDE Laws of Chess taking effect from 1 January 2023 /

Yeah. If we imagine a version of chess (perhaps closer to the original version) in which the King does get taken to end the game, and “check” does not force a response, then in the scenario @3AxisCtrl lays out, if White fails to move his King or take the checking knight, then Black takes the White King before anything else can happen, and Black wins. The modern rules of chess just force the players to avoid inadvertant loss of a King, but the basic logic of taking turns still apply

Like Duck Chess!