Yes, it is sufficient. Your question actually touches on a more general problem. A proper class, generally, is not an object. Hence $V$ is not an object, $j$ is not an object, and if $M$ is a proper class, $M$ is also not an object. So, regardless of our philosophical inclinations, how can we refer to those things?
The trick is, one can assume that classes $M$ and $j$ are definable from parameters by $\Sigma_2$ formulas. Moreover, the assertion that $j$ is an elementary embedding, with exception to the case where $j \colon V \to V$, is equivalent to the assertion that for all formulas $\varphi(x)$, all ordinals $\alpha$, and all sets $a \in V_\alpha$ $$V_\alpha \models \varphi[a]$$ if and only if $j(V_\alpha) \models \varphi[j(a)]$.
So, getting back to your original question, the answer is yes. One can talk about an elementary embedding $j \colon V \to M$ in terms of incomplete parts of $V$, that is in terms of $V_\alpha$.