Unlike en-passant and castling, pinning and discovered checks are consequences of lower-level rules.

At the "Is this move legal?" level, they don't need unique rules of its own if the lower-level rules are specified correctly.

That rule doesn't mean it's illegal to move a piece that's pinned. It just means that it's illegal to move it to a square that would expose the king. For example a pawn that's pinned vertically can still push forward, it just can't capture diagonally.

That's why treating colloquial concepts like "pinning" as though they are rules in and of themselves is not really precise or productive.

You can also pin a pawn to a queen, but the pawn can still legally move.

The point is that, logically, the first part of that rule (“expose the king”) is implied by the second part (“leave that king”), so the first part is redundant. You could simplify the rule to:

No piece can be moved that will leave the king of the same color in check.

Pinning isn’t a rule, it’s just something that arises from other rules.

Also, pinning can happen with pieces that don’t include a king, which means you can just move out of the pin and expose whatever other piece.

It’s just a chess tactic, not a rule. It’s like saying a chess skewer is a rule too.