Which types of first-order signatures have proper pseudo-elementary classes?

Question

Let $L$ be a first-order signature. I define a class $K$ of $L$-structures to be proper pseudo-elementary if it is a pseudo-elementary class which is not an elementary class. I believe, (and correct me if I am wrong), that some signatures $L$ have no proper pseudo-elementary classes. That is, every pseudo-elementary class $K$ of certain signatures $L$ is already an elementary class. For example, if $L$ is the empty signature of pure equality, there are no proper pseudo-elementary classes. I conjecture that the only first-order signatures which do not have proper pseudo-elementary classes are the empty signature and signatures which only contain constants, with no relation symbols or function symbols, and all other signatures do have proper pseudo-elementary classes. Is this conjecture true?

Answer 1

In fact every signature has proper pseudo-elementary classes.

Let $\kappa$ be an uncountable cardinal which is larger than $|L|$. Then the class of $L$-structures of size $\geq \kappa$ is not elementary (by downward Löwenheim-Skolem) but it is pseudo-elementary: add $\kappa$-many new constant symbols and consider the theory $T$ asserting that they are all distinct. Then a structure $A$ has size $\geq\kappa$ if and only if it is a reduct of a model of $T$.