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Send_in_the_Killer_Clowns
I would actually agree that one must be a Necromancer and Immortal to be a Lich, but I use Phylacteries as the prime requirement because I see that as combining those requirements, and also providing a way of making them distinct from the other methods of achieving Immortality you mention. For instance, a Necromancer who achieves Immortality by willingly infecting themselves with Vampirism would become a Vampire, not a Lich, despite in both cases they use Necromancy to achieve Immortality.
In my experience, Creating a Phylactery is usually considered a part of Necromancy, both because it deals with Souls and Death Magic, and often requires the sacrifice of a Life, such as in Oldy Voldy's case. Though I could easily see a Magic System where that isn't the case, where working with Souls, even removing them to create a Phylactery, does not in any way involve what we generally consider Necromancy, or any magic that directly relates to Death. That could be an interesting system, because I would wonder if Magic that manipulates the Soul would still have the negative associate that Necromancy usually has.
Now the potential Spanner in my definition I've just thought of are Golems, which usually in modern Fantasy are created by removing the Soul of a Living Thing and putting them into a Construct. Would the count as a Lich? I think it wouldn't because the Seat of the Soul is not Separate from the Entity the Soul is Controlling, which I suppose I would consider to be Part of the Definition of a Phylactery. Alternatively, Golems usually just use the Soul as a Power Source, and don't allow it to assert its will, so that's another potential distinction.
Interesting Thoughts. Ultimately I could see Liches having other restrictions for different systems, and I won't try to impose the Phylactery Requirement on a setting that wanted to do something different.