I mean that neither nature nor theory requires one to choose Minkowski as the background spacetime, although it is the most popular choice in perturbative General Relativity.

I've been avoiding sign conventions and coordinates deliberately; I think they would be more distracting here, and of course the choices made have no physical implications.

(Using \eta without it necessarily meaning Minkowski spacetime is arguably an abuse of notation, if that's what you mean. :D)