What is a minimum vertex separator as in this definition?
It seems that $Sep(u, v)$ is a vertex separator extended to graphs where $u$ and $v$ are adjacent, by ignoring their common edge.

I assume the correct definition is  


  ... vertex separator in $G$ if $\{u,v\} \notin E$ or in $G′=(V,E \setminus \{\{u,v\}\})$ otherwise.


As given, the definition makes no sense. There is no point in removing $\{u, v\}$ from $E$ if $\{u, v\} \notin E$ and, of course, $\{u, v\} \not\subset E$.

In this case, $Sep(u, v)$ can be $\emptyset$ if $\{u, v\}$ is the only edge connecting two components, or if the graph had two components to begin with.