> The section says "models" but it should work for syntax too not just semantics (the syntax is the "initial" semantics).

Can you elaborate on that?

Do you mean initial in terms of initial objects? [0]

Is this related to Herbrand universes? [1] (in which we build an interpretation of first order logic where each symbol stands as themselves)

[0] https://en.wikipedia.org/wiki/Initial_and_terminal_objects

[1] https://en.wikipedia.org/wiki/Herbrand_structure