(Everything is assuming $V=L$.)

Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ and let $$E_\kappa^+=\{\mu<\kappa: \mbox{$\exists$ an elementary substructure of $(L_{\kappa^+}, L_\kappa)$ which is isomorphic to $(L_\kappa, L_\mu)$}\}.$$ (In each definition $\mu$ ranges over ordinals.) Here "$(L_\alpha, L_\beta)$" denotes the structure $(L_\alpha; \in)$ augmented by a predicate for $L_\beta$.

Clearly $E_\kappa^+\subseteq E_\kappa$. In the course of a problem I'm working on, I assumed that $E_\kappa$ and $E_\kappa^+$ are quite different, but thinking about it in more detail this seems unjustified. My question is:

Is $E_\kappa^+$ necessarily a proper subset of $E_\kappa$?

I would also be interested in specific cases, e.g. $\kappa=\omega_1$.

EDIT: Oh dear, this is what I get for doing math late at night: the question above has a terrible typo - $E_\kappa^+$ is supposed to be the set of $\mu$ such that *for some $\alpha<\kappa$* we have $(L_\alpha,L_\mu)$ is isomorphic to an elementary substructure of $(L_{\kappa^+}, L_\kappa)$. This completely changes the meaning of the question, of course.

I'm leaving the question as is and accepting Joel's answer, since it is a good answer; but I've asked a corrected version of the question at Levels of L resembling each other, take 2.

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