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Originally Posted by George
The point is to focus on the importance of a changing vase as a causal action.
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Right, so we can understand and predict the motion of the Earth without needing to specify if it orbits the Sun or if the Sun orbits the Earth, but we will need to change our answer if someone swaps in a slightly larger Sun far more than we would need to if someone swapped in a slightly larger Earth! Taking that as evidence that the Earth orbits the Sun is to adopt a slightly different meaning for "orbit", that the causal agent "deserves" to occupy a special place in the coordinates. That is often done, just as we say the Earth spins rather than saying the stars go around us once a day. But the equations we use as "laws of nature" don't know the difference-- we view it as quite fundamental that those equations are coordinate independent.
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This is better seen, I think, when we regress to the antipodal apples scenario where apples on opposite ends of the globe fall simultaneously toward the ground so a fixed Earth model (relative to the apples at least) is superior than any other model for human semi-reality consumption. 
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But again the "specialness" of the Earth in that scenario stems entirely from the fact that it is the source of the gravity, so the action of the gravity respects the location of the Earth. If that is used to justify choosing coordinates that also respect that location, fine, but there is no automatic need to choose a coordinate that is built around the gravitational source. Coordinates are all about
convenience, not
philosophy. There is no more fundamental statement of what "relativity" means.