A question about second-order logic and inaccessible cardinals.

Question

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models T$ iff $X \models_{V_\kappa} T$, where $\models_{V_\kappa}$ is the relation on $V_\kappa$ obtained by relativizing the definition of satisfaction to $V_\kappa$ (not by restricting $\models$ to $V_\kappa$).

Question. Let $M$ denote a transitive model of ZFC. Suppose that for every second order theory $T \in M$ and every structure $X \in M$, we have that $X \models T$ iff $X \models_M T$, where $\models_{M}$ is the relation on $M$ obtained by relativizing the definition of satisfaction to $M$. Is $M$ necessarily equal to $V_\kappa$ for some inaccessible $\kappa$?

Answer 1

No, not at all. But it does tell you something about $M$. It tells you that it knows about sufficiently many of the subsets of $X$.

So if, for example, $M=V_\kappa$ for a worldly cardinal which is not inaccessible, then this is also true. For obvious reasons: if $X\in M$ then $\mathcal P(X)\in M$, so every possible subset is in $M$ and $M$ and $V$ agree about second-order (and even a bit higher than that) statement regarding $X$.