You don't seem to understand that those Jewish traditions are far newer than our Christian ones. Lilith isn't in the Bible, or in Christian Tradition.
Maybe in Jewish tradition, but I still don't see the relevance if Lilith existed or not, what does that matter? As I recall, Lilith was a bad seed which was why she was allegedly cast out of Eden first and Eve was created.
Lilith isn't real according to us, buddy.
Lilith is a later Jewish tradition, the Christians never believed in it. And unless you showed us where in the Christian Tradition this story is authenticated, then we won't accept it, no matter how much you insist she existed.