To make this more readable, I've put some definitions/conventions at the end of this question.
I'm interested in when a given interpretation isn't "missing any information." Specifically, say that an interpretation $\Phi$ in a theory $T$ is optimal iff for every extension $\Psi$, every $\Sigma_\Phi$-structure $\mathcal{A}$ satisfying everything $T$ proves about $\Phi$ can be expanded to a $\Sigma_\Psi$-structure $\mathcal{A}'$ satisfying everything $T$ proves about $\Psi$. And say that an interpretation is weakly optimal if the above holds restricted to countable structures. (Relating this to an earlier question of mine, an optimal interpretation is "relatively sharp," except now I'm explicitly disallowing "infinitely long" interpretations - see the end of this question.)
Example: the usual interpretation of $\mathsf{ACF_0}$ inside $\mathsf{ZFC}$.
Weak example(?): I believe that the usual interpretation of $\mathsf{PA}$ inside $\mathsf{ZFC}$ is weakly optimal, due to the role of recursive saturation. It's not clear to me whether it is optimal, however.
Non-example: the obvious interpretation of "the naturals with only the same-parity relation" in $\mathsf{ZFC}$ is not optimal (consider a pair of infinite sets with different cardinalities). However, this interpretation does have an optimal extension, e.g. by adding the successor function.
On a very general level, I'd like to know which theories $T$ have the property that every interpretation in $T$ can be extended to an optimal (or weakly optimal) interpretation - call this "(weak) optimizability" of $T$. I suspect that's too broad to actually be answered, so here's a more tractable version of that question:
Question 1: Is every first-order theory interpretable in a (weakly) optimizable theory?
More locally, I'm curious whether the usual interpretation of arithmetic in set theory is optimal (and not just weakly optimal):
Question 2: Is the usual interpretation of arithmetic in set theory optimal?
Basic conventions/definitions
All languages are finite, and all theories are first-order. An interpretation in a first-order theory $T$ will be a tuple $(\delta,\eta,F)$ where (for some $n$) $\delta$ is an $n$-ary formula, $\eta$ is a $2n$-ary formula which $T$ proves defines an equivalence relation on $\delta$, and $F$ is a finite set of formulas such that each $\varphi\in F$ has arity $kn$ for some $k$ and is $T$-provably well-defined on $\delta/\eta$. An interpretation $\Phi$ extends an interpretation $\Psi$ iff the $\delta$- and $\eta$-parts of $\Phi$ and $\Psi$ are the same and the $F$-part of $\Phi$ contains the $F$-part of $\Psi$. Given an interpretation $\Phi$, let $\Sigma_\Phi$ be the language consisting of a relation symbol for each formula in the $F$-part of $\Phi$ (so $\Phi$ interprets a $\Sigma_\Phi$-theory in $T$).
This is all a bit brief, but it's hopefully clear what's meant. I'm happy to expand on any of it!