Assume ZFC. Let $n$ be a meta-integer. Then  $\kappa$ is $\mathcal{L}^2_n$-correct iff $\kappa$ is $\Sigma_{n+2}$-reflecting, i.e. $V_\kappa\preccurlyeq_{n+2}V$.

Proof: For $n=0$, i.e. 2nd order logic, we have: If $V_\kappa\preccurlyeq_2 V$ then easily $\kappa$ is $\mathcal{L}^2_0$-correct. Suppose now that $\kappa$ is $\mathcal{L}^2_0$-correct. Now for each $\alpha<\kappa$, if $|V_\alpha|<\kappa$ then $|V_{\alpha+1}|<\kappa$, as is easily seen using $\mathcal{L}^2_0$-correctness. So there is a limit ordinal $\eta\leq\kappa$ such that $|V_\eta|=\kappa$. Now observe that $V_\eta\preceq_2 V$ by using $\mathcal{L}^2_0$-correctness. It follows that $\eta=\kappa$ (if $\eta<\kappa$, take $\alpha<\eta$ such that $\eta\leq\beta=|V_\alpha|<\kappa$, and using $V_\eta\preceq_2 V$, show $\beta<\eta$ for a contradiction). So $V_\kappa\preceq_2 V$ as desired.

For $n=1$: Suppose $V_\kappa\preceq_3V$. Let $T\in V_\kappa$ be a satisfiable $\mathcal{L}^2_1$-theory. Then there is an ordinal $\eta$ such that $V_\eta\preccurlyeq_2 V$ and $M\in V_\eta$ such that $V_\eta\models$"$M\models T$", where the latter statement uses $V_\eta$'s own notions to compute $\Sigma_2$-elementarity, and hence the $C_0$ quantifier. But this is a $\Sigma_3$ statement, so $V_\kappa$ models it, and note that this really gives a true model of $T\in V_\kappa$.

Conversely, suppose $\kappa$ is $\mathcal{L}^2_1$-correct. So in particular it is $\mathcal{L}^2_0$-correct, and hence $V_\kappa\preccurlyeq_2V$. We want to see $V_\kappa\preccurlyeq_3V$. Let $x\in V_\kappa$ and suppose $V\models\exists w\varphi(x,w)$ where $\varphi$ is $\Pi_2$. Let $\alpha<\kappa$ with $x\in V_\alpha$. Consider the $\mathscr{L}^2_0$-theory theory $T$ in parameters in $V_\alpha$, which describes a rank segment $V_\eta$ of $V$ such that $\eta$ is a cardinal and $V_\eta\preccurlyeq_2V$ (by using the $C_0$-quantifier to require $\eta$ to be $\mathcal{L}^2_0$-correct), and $V_\eta\models\exists w\varphi(w,x)$. Note this is satisfiable, and hence satisfiable in cardinality ${<\kappa}$, and hence $V_\kappa\models\exists w\varphi(w,x)$.

Now proceed in this manner.

So in particular, least $\mathcal{L}^2_0$-correct cardinal is $<$ least strong
and least supercompact, these are $<$
least $\mathcal{L}^2_1$-correct  $<$ least extendible
< least $\mathcal{L}^2_2$-correct.