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