In the Bible, they are mentioned separately. Nowhere did it say that they were one and the same entity, even if they perform similar functions. Same for Beelzebub. However it has been years since I read it; I might have forgotten something.
From a pre-Abrahamic perspective, they are completely different. Lucifer was once a God of light, identified with Venus/the morning star. The Shayton/Satans were an especially powerful race of djinn 🧞. Baal Zevuv, which means "Lord of the Lofty Abode" not "Lord of the Flies," was a God worshipped in what is now Palestine.
By the way, Mephistopheles is not in the Bible, as far as I remember.