[pdonis]

In cases where the integral you describe is well-defined and physically meaningful, yes, you are correct, it is a center of mass (or center of momentum) integral not a geometric center. But it's not the center of mass of the "volume" over which the integral is done, it's the center of mass of the stress-energy over which the integral is done. In order to obtain the function m(x), you need to look at the stress-energy tensor.

Also, the integral you describe will not, in general, be invariant; it will depend on your choice of coordinates, because you are integrating over a spacelike surface of constant coordinate time, and which surfaces those are depends on your choice of coordinates.

Your intuition about the "usual" invariant mass is based on the special cases where the integral you describe can be equated to one of the known invariants, the ADM mass, the Bondi mass, or the Komar mass. (Strictly speaking, even the Komar case is problematic, because the integral in question in a general stationary spacetime does not necessarily converge. In cases where it does converge, AFAIK the spacetime must be asymptotically flat and the Komar mass is equal to the ADM mass.) But an expanding universe is not one of those cases.