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?
The point is, the "Killer King" achievement is achieved by moving the king and delivering checkmate.