I don’t think I need to tell you that the appearance of the word “Lucifer” in the Bible referred to a king named Nebuchadnezzar, not the Devil. Lucifer, as well as Nocturnus, Helel, Noctifer, etc. are just some of the many titles given to the “Morning Star” (which turned out to be planet Venus). The comparison of Nebuchadnezzar to that “star” was simply an illustration of his former glory: “Nebuchadnezzar, you were once the bomb dot com, but then you fucked up.” Ergo, the oral tradition of an Angel named Lucifer attaining the title of Satan after He rebelled is simply neo-Christian folklore. The Islamic equivalent of this tradition gives Iblis/Shaitan the name “Azazel” before His rebellion. Nonetheless, it is very clear to me that the title Lucifer is applicable to Satan.
You may be aware that “ha-Satan” used to be a common noun, as opposed to a name signifying one particular entity. It just means “the Adversary” or “the Accuser.” However, the title of Satan was often applied to the Prince, and since it fits rather nicely, we use it.
Well, as it turns out, the title of Lucifer was common enough as well. We already see that it was attributed to Nebuchadnezzar, the epic poem Metamorphosis attributed it to another [minor] Roman deity, and in the Vulgate’s book of Revelations, Jesus Christ calls himself Lucifer. The Greek words for “light-bringer” (Phosphoros or Phaesporia) have been applied to Hekate and Artemis. The Egyptian word for “light-bringer” (which is “Khut”) has been applied to Isis. Furthermore, there is more than one deity associated with the morning “star”: Isis and Tara share this identification. Ergo, the title of Light-Bearer is applicable to any deity which warrants the moniker.
Since the title of Lucifer is common, it can easily be bestowed upon any deity that is worthy. Since Satan is a harbinger of free thought, Lucifer is an apt title for him to bear. Ergo, I say that Lucifer is sometimes Satan, depending upon the intent. Satan is pretty used to responding to random names anyways– there are Satanists who consider Him to be Enki, some who call Him Marduk, and some who call Him Tiamat. If Satan responds to these monikers, Satan will certainly respond to Lucifer.
In Stregheria, or generational Italian witchcraft, the Devil is referred to as Lucifer, Lucifero, and Lucibello (and more). In this religion, whose practitioners sometimes work with Cain, Lucifer is the Brother of Diana, as well as a Spirit who rebelled against Yahweh. It is far more reasonable to adopt the titles used by Italian witches than it is to adopt titles used by Sumerian clergy.