I think Lilith is a very pretty name although knowing the meaning, I might take a step back from using it. We have religious friends and family though so baby Lilith and baby Lucifer would maybe cause us more problems than it's worth/it's not the hill I'm willing to die on.
For me with Lucifer, it's knowing that it is so well known in relation to the Devil and people sometimes use it to refer to the Devil. I like the idea of it as a fallen-angel name in a way, but not in a directly referencing Satan himself way, if that makes sense? Cool for a cat though
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The baby Lilith I know is gorgeous. Her parents are quite alternative so I can see why the name would be both beautiful and have cool/edge appeal for them but I certainly don't think they've used it in a negative way/chosen it to imply negative things about their baby. It's less obvious than Lucifer for a boy, too - I had to look it up when I read courtiers message, as I had no idea it had such a background. I just thought it was a pretty name.