The point is, the "Killer King" achievement is achieved by moving the king and delivering checkmate.
Yes, and they are not separate things. The king is only called "killer king" because it killed (checkmated) the other king by its action.
Of course it is a simplification to assign the checkmating act to a single unit It comes from the insistence on nominating one target for admiration or praise. Truth is that a king is checkmated by the complete board position with all the present chess units which are willfully substituted by a single deliverer of the end result. As they hand a music reward to the band leader who represents the whole band but in no way contributes more to it than the other members.
Like I said, then try it. If you believe the king can attack the square the enemy king occupies, do it. Play in a tournament, use your king to attack the square the enemy king occupies, and see what happens.
I notice you did not answer the question though. In our example, which piece is giving check, the pawn or the bishop?