For s set $S\subseteq RE$, so call feature of language $S=\emptyset$ vs. $S=\{\emptyset\}$
Assume that all languages are over the alphabet $\Sigma$. What you have here is a bit of ambiguity in the meaning of $\emptyset$ (recall that the emptyset is defined w.r.t a universal set, and here $\emptyset$ is used w.r.t different universal sets). Indeed, $S = \{ \emptyset\}$ refers the set of languages containing only the empty language, that is,  in this case, $\emptyset\subseteq \Sigma^*$. Also, $S = \emptyset$ refers to the empty set of languages, that is, in this case $\emptyset\subseteq 2^{\Sigma^*}$.
As you noted, if $S = \emptyset$, then $L_S = \{ \langle M\rangle: L(M)\in \emptyset\} = \emptyset \in \text{R}$. Now if $S = \{ \emptyset\}$, then $L_S = \{ \langle M\rangle: L(M)\in \{\emptyset\}\} = \{ \langle M\rangle: L(M) = \emptyset\} = E_{TM}$ which is known to be in $\text{coRE}\setminus \text{R}$.