How do these additional cases fit into this Theorem about the optimal substructure of a longest common subsequence?
If $x_m = y_n$ then any longest common sequence must end with $x_m = y_n$. This is what the first item covers.
If $x_m \neq y_n$, then there are three types of longest common sequences:

Longest common sequences ending in $x_m$.
Longest common sequences ending in $y_n$.
Longest common sequences ending in neither.

These are all the cases. There is no other case.