I hang around that Catholic Answers Forum, because it occasionally produces something interesting. My usual zone is the 'Non-Catholic Religions' forum. Someone asked whether anyone knew the meaning of the various kinds of modern Satanism, giving this list:
LaVeyan Satanism; Temple of Set; Modern Satanism; Theistic Satanism; Luciferianism; Satanism Dabbling
This produced a somewhat anemic discussion, with the usual fear-stricken handwringing by the stereotypical RCC types. I was, eventually, moved to comment and, hating to waste a few minutes of writing, will post it here. I guess this comes under history and taxonomy of modern occult religions...
I find the original question on this thread interesting, mainly because it illustrates that Satanism is actually now a growing thing, with enough people who care about it to actually discuss what their particular path is called. That was never the case in the past, as far as I can see from history.
Medieval Satanism is mainly a thing of Christian legend, a pathology of the medieval church that had little or no expression among the people. To my knowledge there is no single example of a book or artifact of 'devil worship' or 'Luciferianism' from the middle ages. There are the transcripts of the trials, which may have a few remnants of a folk identification of 'the devil' with one or another of Europe's pre-Christian Gods. However the prejudices of the monks who recorded the statements must be weighed as well. The occult tradition of the late middle ages gives occasional rites of offering to spirits angelic or demonic, but those are purely practical formulae, usually followed by prayers to the usual Christian God. In no place have we found an example of a ritual of satanic worship, or a satanic chapel, or statue, or liturgical item, from the medieval or rennaissance eras.
In the Renaissance we find a very short list of tales of people who actively perverted the Christian sacraments for personal 'magical' goals, so there had been a problem with church functionaries misusing their spiritual authority - that still isn't Satanism... Apparently the use of Christian rites for magical goals had formerly been common. This from wikipedia, but it's not wrong: "Additionally, the Rite of the Mass was not completely fixed, and there was a place at the end of the Offertory for the Secret prayers, when the priest could insert private prayers for various personal needs. These practices became especially prevalent in France. ... the institution of the Low Mass became quite common, where priests would hire their services out to perform various Masses for the needs of their clients - such as blessing crops or cattle, achieving success in some enterprise, obtaining love, or cursing enemies (one way this latter was done was by inserting the enemy's name in a Mass for the dead, accompanied by burying an image of the enemy). Such practices were eventually condemned by the Church, however."
In the 19th century we start to see what we would call 'black masses' performed as tourist gags in france, and the romance of 'black magic' was culturally established. Europe had at least a couple of Luciferian magical groups in the late 19th and early 20th century, even as most occultism continued the search for the divine, as it has always done. The 19th century saw the invention of the idea of 'left and right hand' paths, and the creation of the famous 'baphomet' image by the former RCC priest Eliphas Levi, based on some remnants of templar symbolism. Levi wasn't a Satanist - he invented the Baphomet to embellish his stories of 'Satanic magicians'.
Many call Aleister Crowley the father of modern Satanism, but I just don't think it's true. Crowley remained an earnest seeker after divine truth throughout his life, though his search took him in strange directions. He *did* have an attitude about public orthodox Christianity, though he valued the insights of Christian mystics. He enjoyed playing with the images from John's Apocalypse, and he made them key to his new mythic system - that was, and is, sure to bristle many Christians.
Self-identified 'Satanism' barely existed before Lavey. After Lavey it caught on with a few folks, and they quickly developed schisms, as new religions will do. All the categories that the original poster listed simply did not exist in, say 1985. I think he missed a couple of categories, such as Daemonolatry, but never mind. Some Satanists attempt to find a pre-Christian Satanism in one or another ancient Pagan religion, but the whole effort is pretty flimsy - there just isn't a category for the Miltonian 'Rebel Prince' or the 'author of evil' in most traditional mythologies.
So, my position is that the old Christian notion of secret conspiracies of demon-servers working against society is nonsense, and so is the Satanist notion of secret ancient cadres of brave spiritual adventurers. Satanism is mainly a modern invention, growing mainly from the imagination of the medieval church, and only secondarily (or less) a part of 'occultism' or 'Paganism' or even 'witchcraft'. Mostly there's so little of it that it doesn't amount to anything but a body of writing.