Unclear about proof for unique MST given graph G with distinct weights
The edge $e$ is not necessarily the cheapest edge of $G$ that is in $T$. Rather, it is the cheapest edge in the symmetric difference $T \triangle T'$. It could belong to either $T$ or $T'$ (but not both!), but since both cases are the same, we assume that it belongs to $T$ without loss of generality.

We don't need to separately consider the case that $T$ is disjoint from $T'$ and the case that they share some edges. All we need to know is that they are different and so their symmetric difference is not empty.