The Pauli principle is not a force in the sense that gravity is not force. Yes, you can distinguish between a "force" and the phenomenon responsible for the force (gravity vs gravitational force). What is the difference between these two statements?

1) That's not a force, it's the playing out of the fundamental symmetries in quantum physics, normally phrased here as the Pauli exclusion principle.

2) There's no force on that falling object in a vacuum, it's just following the geodesic dictated by the symmetries in General Relativity.