A given theory $T$ can interpret the language of arithmetic in different ways, and therefore it seems to me like it is possible that in one of these ways $T$ would be $\omega$-consistent but not in the other.
Assuming PA is $\omega$-consistent, there seems to be an easy way to construct such a theory: Let $T$ have two sorts and put the axioms of PA (Peano Arithmetic) for one of the sorts and PA+not(Con(PA)) for the other. Then depending on which sort we use to interpret arithmetic we would either get something $\omega$-consistent or not.
Is this correct? If so, $\omega$-consistency seems like an ill-defined concept. How do we deal with this issue? A similar question can arise if we have some model of some theory and want to know whether $Con(T)$ for some $T$ is true for that model, as the answer depends on the encoding.