Test all three by the same standard:
Genuine: both true impossible; both false impossible. Example: an integer is even or odd.
Exclusive of conjunction: both true impossible; both false possible. Example: an object is a tree or a stone.
Exclusive of omission: both false impossible; both true possible. Example: not human or animal.
The genuine form combines both exclusions. The remaining kinds exclude only one of the two combinations. Ask “Can both be true?” and “Can both be false?” before deciding.