Let $\mathcal{L}$ be a logic and $\mathscr{K}$ a class of structures in the vocabulary of $\mathcal{L}$. We say that $\mathscr{K}$ is a (basic) elementary class iff there is $\phi \in \mathcal{L}$ such that $\mathscr{K} = \{\mathfrak{A} \; | \; \mathfrak{A} \models \phi\}$. $\mathscr{K}$ is a pseudo-elementary class (or, alternatively, a projective class) iff there is an elementary class $\mathscr{K}'$ in a vocabulary $\tau' \supseteq \tau$ (where $\tau$ is the vocabulary of $\mathscr{K}$) such that $\mathscr{K} = \{\mathfrak{A}\restriction \tau \; | \; \mathfrak{A} \in \mathscr{K}'\}$. Finally, a class $\mathscr{K}$ is a relativized pseudo-elementary class iff there is $\tau' \supseteq \tau$, a unary relation symbol $U \in \tau' \setminus \tau$, and an elementary class $\mathscr{K}'$ of vocabulary $\tau'$ such that $\mathscr{K} = \{(\mathfrak{A} \restriction \tau)\mid U^\mathfrak{A} \; | \; \mathfrak{A} \in \mathscr{K}' \text{ and } U^\mathfrak{A} \text{ is } \tau\text{-closed in } \mathfrak{A}\}$. $(\mathfrak{A} \restriction \tau)\mid U^\mathfrak{A}$ is the relativized reduct of $\mathfrak{A}$ (basically, you consider only the domain "U" of $\mathfrak{A}$).
My question is rather simple: what are the inclusion relations between these classes? It seems to me that $\mathrm{EC}_\mathcal{L} \subseteq \mathrm{PC}_\mathcal{L} \subseteq \mathrm{RPC}_\mathcal{L}$. My reasoning is that we can always restrict the vocabulary of the structures in a $\mathrm{EC}_\mathcal{L}$ class to the basic vocabulary of the $\phi$ in question, and that, if the logic allows sentences such as $\forall x Ux$, then every $\mathrm{PC}_\mathcal{L}$ can be made into a $\mathrm{RPC}_\mathcal{L}$ class; is this correct?
EDIT: Added a more explicit definition of relativized pseudo-elementary class.