Earlier quoted context omitted.
Newton mechanics was wrong: because it didn’t account for the relativistic terms, all the equations are off by some amount. It turns out that we can ignore the error in practical contexts, but that’s still being “wrong”. Another example: the math of a geocentric solar system is often precise enough for practical purposes and so, when it’s simpler, it can be useful to use it instead of Newtonian or relativistic models…
I would say Newtonian mechanics is a good approximation (as it is accurate in its domain, plus easier to conceptialize and compute with than either QM or GR). Approximations aren't "wrong", they are just simplifications of something less wrong, usually in certain corner cases. Sometimes approximations are derived from the broader theory, and sometimes the broader theory comes later. (And of course, sometimes old theo…
I think it’s true that wrongness is relative to the use-case. But, on the other hand, before things like the Ultraviolet Catastrophe and the Michelson-Morley experiment, a lot of physicists thought they had the big picture of the universe basically right and just had to fill in the details. The revolution of GR and QM was barely even suspected and this should give us pause when it comes to our confidence in the modern picture of the universe. Relativity and QM will always be useful as models of the spheres they model, but the worldview we’ve built on top of them could change drastically overnight (in historical terms).