I contend that Lucifer is primarily a phenomenon that can be found within all beings, including perhaps oneself.
And that the word “Lucifer” is only a name with a certain archetypal energy attached, and not directly linked to the historical term in any way (I’ll link below some info from one of our more scholarly members on this).
As for when Luficer appears to take on personality traits, with preferences and so on, literally everything does that, from low-level servitors onwards, and it can be tested by anyone willing to use this method:
Your entirely and intentionally imaginary friends will start becoming more “real” and developing their own personas, and eventually some might also link to known beings, such as major demons and clasical deities. 
I think if someone goes looking for certain qualities in a being, they’ll find them; the ancient Egyptians used to syncretise their deities and that often happened at times of major social changes, when they needed a specific thing on a deep emotional level.
This is why IMO it’s not worth having any major angst over, if “your” Amaymon says he’s definitely NOT Lucifer, when someone else says he is.
Academic stuff with sauce:
And my own experience described in a reply here: Why Is The Sigil For Lucifer The Same As Satan? - #3 by Lady_Eva