In a real situation we cannot know what the "true" model is and I'd even say there is none. ("All models are wrong but some are useful" said George Box.)
A true model is always a model that is assumed to be true (in some kind of artificial model world) in order to investigate mathematically what happens in case this is true.
In standard statistical thinking, there exists a "true" model, and statistical methods attempt to fit or estimate or select it (in the simplest and most standard case it is assumed that the model that we attempt to fit is the true model). This, however, is a "useful fiction". It makes sense to distinguish the unknown real truth from an estimator or fit of it, however a "true model" only exists in the world of mathematics.
In reality all you can do is to reject a certain model if you observe something that should hardly ever happen were the model indeed true. Models can often be motivated from knowledge of the situation, but that doesn't necessarily make them true.
When doing statistical theory and research, which model is assumed as true can be decided by the researcher and depends on what the researcher is interested in.
For example, in your situation with models
$ log(wage) = \beta_0 + \beta_1 S + \varepsilon, $
$ log(wage) = v_0 + v_1 S + v_2 A + v, $
you could be interested what happens if the first model is true and you fit any of the two models (or both and compare them), or analogously what happens if the second model is true. You may have a model selection method and may be interested in whether it selects the true model with high probability, so you can assume either model as true and ask whether this will likely be found (in much theory such situations can be assessed in the same go, so they'd give you an indication what happens if any of the models of interest is true, but this does not always work).
Rather standard are results where you fit a certain model, you assume this model to be true, and then, by theory or simulation, you find out about how good your fit is expected to be.
You may also be interested (as people in robust statistics are) what happens if you try to fit a certain model, and in fact a different model is true.
So any model can be assumed as true, if you are interested in analysing what happens to a certain statistical method in case the model is true. It is the job of the author to explain clearly which model is assumed to be true where, so as a reader you should know, but in principle any model goes, there are no general criteria that a model assumed as true has to fulfill.