It entirely depends on which Lucifer or Satan you’re referring to.
The Roman God Lucifer, God of the Morning Star or the entity discovered by Christian demonology?
The Lucifer referred to in the Bible is a Latin translation of the Hebrew word, Helel, which means morning star, in Isaiah 14:12. However, this Lucifer is a poetic description of the King of Babylon, and not Satan.
The Satan of Christianity and the New Testament is a completely different form the Satan of Judaism and the Old Testament. In the latter, the Satan is a title of the Archangel Samael who has the role of accusing men. In the former, Satan is a being who is in rebellion against God that seeks to bring others into rebellion against others, he is said to be the serpent of the Garden of Eden that decieved Adam and Eve to eat the fruit of the Tree of the Knowledge of Good and Evil.
The connection between Lucifer and Satan was a theological development of the third century, when the Latin Vulgate was created and in a mystical text saw the true King of Babylon to be Satan.
So, people who say that Lucifer isn’t Satan or that they are, are likely referring to separate Satans (while the Old Testament Satan has absolutely nothing to do with Lucifer, the New Testament Satan does).