In not sure if this is the appropriate way to tie into this convo, but I hit many of these points in a thread a while back that’s now locked. But I can regurgitate all of it in here.
But I definitely don’t think Satan is Lucifer since Lucifer is just a weird translation for morning star which is better rendered Venus. Lucifer is strictly a KJV mistranslation in my opinion and it was a poem about the fall of a man and refers to to morning star as a man several times. To tie it to a fallen angel even with the eden references just does not seem to fit systematically throughout scripture as a whole.
In the Torah the only fallen angels are the enochian ones at best, and in the New Testament Satan is never actually in hell being punished but roams earth. Satan was not sent to hades ( the grave ) to be punished, then escaped and came to earth to destroy and then gets locked up and eventually judged and cast into hell ( the lake of fire) and destroyed along with the body and souls of the wicked.
So it makes sense that Satan is not the character in question concerning Lucifer.