How come $REC$ languages are included in $NRE$
I've never seen "NRE" defined to mean "not recursively enumerable", which is the short way of saying what you've defined.  If you take it to mean "not recursively enumerable" then no recursive language is in NRE because every recursive language is recursively enumerable.

The only definition of "NRE" that I've seen is that people in the field of DNA and membrane computing seem to use it for the set of Turing-computable sets of natural numbers, i.e., the sets that can be written as $\{|x|\mid x\in L\}$ for some RE language&nbsp;$L$.