I don't like the term "Model" or "Model theory". It prompts people to think of it like a model airplane or something. Like a smaller, less practical representation of something in the real world. I would prefer "interpretations" and "interpretation theory". Or, alternatively, "structures" and "structure theory".
The word "interpretation" does occur in model theory. It refers situations in which one structure represents or contains another structure. Funnily, in this case, I think the word "model" would be more appropriate because this is more analogous to the idea of a model airplane.