In general relativity, gravitation is not a force, unlike the other fundamental interactions (i.e. electromagnetism and the weak and strong nuclear forces). Gravitation is solely due to the curvature of space-time, which is bent by matter and energy indeed. It is really important to make the distinction, because the fact that gravitation is not a force is precisely what allows to explain the equivalence principle within the framework of general relativity.

Also, for all fundamental interactions, the fields are the fundamental "objects", not their quantum "realization" -- as you seem to imply in your last section. The "graviton" is the hypothetic quanta of gravitation, but gravitation does not need to be in a quantum regime to mediate (or "carry") gravity. You could also have talked about gravitational waves.