Recursively inaccessible ordinals and non locally countable ordinals This answer seems to imply that: for an ordinal $\alpha$, to be recursively inaccessible (i.e. $\alpha$ is admissible and limit of admissible) implies to be not locally countable (i.e. $L_\alpha \models \exists \beta \ ``\beta \text{ is uncountable"}$). Here is the relevant excerpt:

If there is some $r\in L_\alpha$ with $\omega_1^r=\alpha$, then $L_\alpha$ will be locally countable (= $L_\alpha\models$ "every set is countable"). But plenty of countable admissible $\alpha$s don't give rise to locally countable levels of $L$! In particular, if $\alpha$ is an admissible limit of admissibles (= "recursively inaccessible") then every real in $L_\alpha$ is contained in some admissible $L_\beta$ with $\beta<\alpha$.

However I believe that if by "admissible" we read here "$\Sigma_1$-admissible", this statement does not hold as in this sense the first recursively inaccessible ordinal appears way before the first ordinal that is not locally countable (see resp. 2.3 and 2.21 in DA Madore's zoo of ordinals). Whence my question:

Does the statement hold if by "admissible" we read "$\Sigma_n$-admissible for all $n$"? If no, can we describe the first ordinal which is not locally countable in term of higher recursive inaccessibility?

 A: Suppose $L_\alpha\models$"There is a largest cardinal". Then ($L_\alpha$ is $\Sigma_n$-admissible for all $n<\omega$) $\Leftrightarrow$ ($L_\alpha\models$ ZF$^-$) $\Leftrightarrow$ ($L_\alpha$ does not project ${<\alpha}$) $\Leftrightarrow$ ($\alpha$ is a cardinal in $L_{\alpha+1}$). So the least $\beta$ such that $L_\beta\models$"$\omega_1$ exists" is $\beta=\alpha+1$ where $\alpha=\omega_1^{L_\beta}$ is the least ordinal such that $L_\alpha$ is $\Sigma_n$-admissible for all $n<\omega$, hence the least such that $L_\alpha\models$ ZF$^-$.
However, if $\alpha$ is least such that ($L_\alpha\models$ ZF$^-$  and $\alpha$ is a limit of $\beta$'s such that $L_\beta\models$ ZF$^-$), then $L_\alpha\models$"Every set is countable". (For each of those $\beta$'s, $L_\beta$ is pointwise definable, so $\beta$ is countable in $L_{\beta+2}$).
If we let $\alpha$ be least such that $L_\alpha\models$" KP + $\omega_1$ exists", then $\omega_1^{L_\alpha}$ is a limit of limits of ZF$^-$ levels, a limit of limits of limits of ZF$^-$ levels, etc.
