No, I was saying that they are degrees of freedom that don't tell you about the geometry, at least not considered by themselves. You can make them change arbitrarily without changing the underlying geometry, by changing coordinates.either these metric components are degrees of freedom, that are allowed to vary (constrained by equations of motion), or they are not. At least some are not. I thought you were telling me none of them are.
That's what my analysis a while back, with the 150 total degrees of freedom of which 130 are "used up" by choosing coordinates, was trying to get at. But I agree there is more that could be said there.what is the relationship between all these degrees of freedom, and your choice of coordinates.
If by "globally" you mean "at every event in the spacetime", then I think I agree that that's enough information to determine the Weyl tensor. I'll have to consider some more to be sure, though.I mean, if you know the mass distribution globally, i.e. in the Schwarzschild metric you know the mass of the source.
I'll defer comment on the rest of your post until I've digested your questions some more.