When I said "state of motion", I was talking about whether motion is inertial or non-inertial. This is indeed frame-independent in general relativity. There are privileged states of motion.

In Newtonian mechanics and the special theory of relativity, the way they are usually formulated, inertial motion is just motion that is non-accelerated relative to an inertial frame. So privileging inertial motion amounts to privileging a set of frames - the inertial frames. Both of these theories are ordinarily formulated so that their equations of motion only hold in inertial frames.

General relativity, in its ordinary formulation, is generally covariant. There is no set of privileged reference frames such that the equations only hold in those frames. Incidentally, this amounts to more than just the fact that spacetime is represented by a Lorentzian manifold (also, your definition of Lorentzian manifold is incorrect). Spacetime is represented by a Lorentzian manifold in special relativity as well. What makes GR generally covariant is that its kinematical and dynamical equations are tensor equations. Tensors are coordinate-independent objects, so tensor equations are true in all frames of reference (that satisfy certain continuity conditions).

So now the theory doesn't just hold in inertial frames of reference. But it turns out that there is still a privileged notion of inertial motion, one that can be expressed in a coordinate-independent manner. Basically, inertial trajectories are trajectories in freefall under gravity, and these trajectories are just the ones that satisfy the geodesic equation, which is also a tensor equation. So yeah, GR does have privileged states of motion.

It's worth noting that the existence of privileged inertial trajectories does not correspond to the existence of privileged inertial frames that extend across the entire manifold. Suppose we construct a frame that is "adapted" to a particular inertial trajectory, i.e. one in which the object following that trajectory is stationary. If space is variably curved, then relative to this frame, some other free-fall trajectory will be accelerated. So, unlike Newtonian mechanics or the special theory, we don't have frames where all inertial motion is unaccelerated.

As for what Eliezer was suggesting, perhaps I have misinterpreted him, but in his discussion of GR he brings up the principle of equivalence a lot. He says, for instance:

This meant you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.

Coupled with his subsequent claim that epiphenomenal distinctions are, as a rule, illusory, he seems to be strongly suggesting that there is in fact no difference between these two states of affairs, which would imply that there is no objective fact of the matter about whether spacetime is flat or curved. This is the claim I was disputing.