I am glad to see that a general question like Is there a relationship between model theory and category theory? receives quite a lot attention and no down-votes for being too general and unspecific. So I feel encouraged to pose some follow-up questions (firstly restricted to first-order model theory).
I hope the following statements are sufficiently sensible, precise and correct.
Each first-order theory $T$ with signature $\sigma$ unambigously defines a class of (ZF-)models.
This class of models of $T$ together with the $\sigma$-homomorphisms form a category (the category of models of $T$).
Two first-order theories with two arbitrary signatures may define equivalent categories of models.
Definition: A first-order theory $T$ provides a model of a category $C$ if the category of models of $T$ is equivalent to $C$.
- Each category $C$ defines a (possibly empty) set of first-order theories: the set of all $T$ which provide a model of $C$.
[Remark: I had to work this question over, since it seemed to be ill-posed.]
Old version: Can infinite concrete categories be specified other than as categories of (ZF-)models of some (possibly higher-order) theory? Examples?
New version (explicitly restricted to first-order theories):
Given an infinite category of models of a first-order theory $T$. Can this category - or one equivalent to it - be specified/represented/given independently of any first-order theory $T$ and its (ZF-)models?
Remark: $T$ of course can be specified/represented/given independently of its models: as a set of formulas.
Why is the notion of models of a (concrete) category so uncommon? (Maybe because the answer to the first question is "No"?)
Is there a genuine model-theoretic notion of two equivalent theories if these have two arbitrary signatures?