Monster Manual 5e - Demilich - p48

Strictly speaking, there are two types of Demilichs.

One kind is the Lich that forgets or fails to feed enough souls to its phylactery and becomes a Demilich (Below).

The other kind is the Lich that tries to become one on purpose like Acererak, who insetted eye gems in his eye sockets that would drain the souls of those who touched them.