Consider below two languages
$L_1=\{<M>|$ M is a turing machine, $M_0$ is a TM that halts on all inputs, and $M_0 \in L(M) \}$
$L_2=\{<M>|$M is a TM, $M_0$ is a TM that halts on all inputs, and $M \in L(M_0)\}$
So, in both languages $M_0$ is a halting Turing machine, so the language of $L(M_0)$ is recursive.
Now in case of $L_1$, it appears to me that $L(M)$ is recursively enumerable, and since recursive languages are proper subset of Recursively Enumerable languages, so $M_0 \in L(M)$ , and since, for the no cases of the language, our turing machine could go into infinite loop, we cannot answer the no case of $L_1$, making $L_1$ Recursively Enumerable but not recursive.
In case of $L_2$, our given machine $M \in L(M_0)$ so it is very sure that the language $L(M)$ should be recursive.
Am I correct in my reasoning.?
PS: I am weak at these RE and REC problems.So, I decided to work on them and discuss my doubts here.None of the above problems are homework problems.