Roughly speaking, say that a logic $\mathcal{L}$ is **self-equivalence-defining** (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ unary relation symbols and an $\mathcal{L}[\Sigma']$-sentence $\eta$ such that the following are equivalent for all $\Sigma$-structures $\mathfrak{A},\mathfrak{B}$:

$\mathfrak{A}\equiv_\mathcal{L}\mathfrak{B}$.

There is a $\Sigma'$-structure $\mathfrak{S}$ such that $\mathfrak{S}\models \eta$, $A^\mathfrak{S}\upharpoonright\Sigma\cong\mathfrak{A}$, and $B^\mathfrak{S}\upharpoonright\Sigma\cong\mathfrak{B}$.

For example, Fraisse showed that $\mathsf{FOL}$ is SED *(this is used crucially in the proof of Lindstrom's theorem - it's a very happy construction)*. That same argument gives as a corollary that $\mathsf{SOL}$ is also SED, roughly because $(i)$ "$X$ is the powerset of $Y$" is second-order expressible and $(ii)$ $\mathsf{SOL}$-elementary equivalence between two structures amounts to $\mathsf{FOL}$-elementary equivalence between their "power-structures."

The nicest logic whose SED status I don't know is $\mathcal{L}_{\omega_1,\omega}$. On the one hand, this logic isn't powerful enough to perform the same sort of "cheat" as $\mathsf{SOL}$. On the other hand, the direct game-theoretic attack analogizing the situation for $\mathsf{FOL}$ results in a game which is a bit too complicated for $\mathcal{L}_{\omega_1,\omega}$ to handle appropriately; see Vaananen/Wang, *An Ehrenfeucht-Fraisse game for $\mathcal{L}_{\omega_1,\omega}$*, and note that this was also an issue in this earlier question of mine. So my question is:

Is $\mathcal{L}_{\omega_1,\omega}$ SED?

I'm separately interested in the general situation of which infinitary logics are SED. It's not hard to show that for $\kappa$ satisfying appropriate large cardinal properties we have that $\mathcal{L}_{\kappa,\omega}$ is SED, but I don't see how to extract large cardinal strength from SEDness.

Note that looking at a single sentence $\eta$, as opposed to a theory $E$, is needed to avoid every (regular) logic being trivially SED; I messed this up in the original version of this question, and this was pointed out by Peter LeFanu Lumsdaine. Separately, I've added the forcing tag since, despite not being part of the question, general principles about forcing turn out to form a key component of the answer.