> Does this mean that gravity is a real form of eneergy and not a simple deformation of space-time?
No, it means that, heuristically, your local definition of "energy" and your local definition of "inertia" are both determined by the geometry of spacetime, i.e., by the same thing. That's why they have to match.
IOW, if gravity were solely a deformation of space-time, then it would not add inertia? Or am I missing or misunderstanding something?