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.
Well, a minor piece can be pinned in front of a queen. It would be legal to move the minor piece in that case, but you'd most likely lose the game if both sides' material were even before the move. Pinning is a tactic, not a rule.
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.
Pedantically I disagree, to leave something in a condition it must have been in that condition in the first place. We could have a game where you're allowed to place your king in check, but if it is in check at the start of your turn you must fix that.
While we're being pedantic though it's not a property of the piece that might be able to be moved that will place the king in check. It's a property of the move. For example we might imagine you have a rook between an enemy rook and your king. You can move the rook along the line between the enemy rook and the king, but not perpendicular to it.
The rule should be:
No move can be made where the moving players king is in check in the resulting position
At the "Is this move legal?" level, they don't need unique rules of its own if the lower-level rules are specified correctly.