I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem to construct an analogue of set-theoretic forcing in Model Theory. This was very interesting to me as it was the first time I had heard this.

I am extremely interested in the $\forall_2$ theory $Art_{k}^{l}$ of local artinian rings of length at most $l$ containing the field $k$ in the language of rings. To get an application, we must insist on the language being countable, which will amount to requiring that the field $k$ is countable.

My question is the following: what are the enforceable models the theory $Art_{k}^{l}$. This means what models $M$ admit the enforceable property of "$M$ is isomorphic to the compiled structure?"

In particular, can we use this to say something about $Art_{\mathbb{Q}}^{l}$ and/or $Art_{\mathbb{F}_q}^{l}$?

Forgive me, the terminology is wonky. I am asking the question because I am not familiar with forcing in model theory much less forcing in set theory. Hodges' "Model Theory" is the only reference I have; however, I know that the omitting types theorem is a fundamental result in model theory. I hope someone is familiar with this construction.