Quote: The association of Lucifer with Satan grew in the early centuries of Christianity, influenced by interpretations of biblical texts that sought to explain the presence of evil in the world. Over time, this led to the popular belief that Lucifer was another name for the Devil, a belief that has persisted despite its shaky scriptural foundation."

Elaine Pagels has a great book on this issue

"The Origin of Satan" as to how the character's status evolved over time from Jewish myth through Christian myth

https://www.penguinrandomhouse.com/books/126050/the-origin-of-satan-by-elaine-pagels/