for some useful definitions of "equal"

But what's the mileage in the claim that you have an approximate reproducion of a moment in time? A battel re-enacment or historical move would cound for some values of "approximate". That's a long way from time travel.

I think the root of your disagreement is an unstated assumption that "Sy follows (in time) Sx" implies "Sy =/= Sx". That is usually true,

If you are saying Sx and Sy are universal states, then you would need a simulator that somehow sacrifices itself to in generating the simulated state.