As for the OP's last question, "Is there a genuine model-theoretic notion of two equivalent theories if these have two arbitrary signatures:"
Sure, it's called [bi-interpretability].1bi-interpretability. See pp. 1378-1379 of this article for a definition of what this means for structures. For theories, we say that T_2 is interpretable in T_1 if there is a family of interpretation formulas (as in the linked definition) such that for any model M of T_1, these formulas define an interpretation in M of some model of T_2. Similarly, as above, we can define what it means for two theories to be bi-interpretable.
If you make the class of models of T into a category by declaring the morphisms to be the elementary embeddings (which seems very natural to me), then it follows directly from the definition I've linked that any two bi-interpretable theories that are bi-interpretable (without parameters) have equivalent categories of models (via the natural "interpretation functors")functors" which translate back and forth between the two languages). .