Here's my essay for the core knowledge of Libertine II for MCoS (numbers 4 and 5).
Different strands of Satanism /Luciferianism and organizations:
"LaVeyan" Satanism - This is the origin of the modern interest in Satanism and represents the thought of Anton Szandor LaVey (Howard Stanton Lavey) as recorded in the Satanic Bible and practiced by the original Church of Satan. Anton LaVey created modern Satanism as a celebration of carnality and as a statement against those hypocritical and anti-human standards found in spiritual white-light religions. "LaVeyan" Satanists do not believe in a literal Satan/devil, but view Satan as a symbol or personification of the human ego as well as the unknown forces of nature. Rituals and magic are used to manipulate reality and other people, as well as a form of psychodrama to purge strong emotions which can impede the will of the Satanic magician or witch. Groups influenced by LaVey tend to have a hierarchical institutional structure with members, Priest/esses, and Magister/Magistra with various levels of authority. "LaVeyan" Satanic groups include the Church of Satan currently led by Magister Gilmore as well as the First Satanic Church led by Karla LaVey. (Note - both the Church of Satan and the First Satanic Church reject the term "LaVeyan Satanism".)
Traditional or theistic Satanism - These are composed of groups which often claim to have ancient roots. They believe in Satan (or Enki or Set) or the demons as literal supernatural beings which may or may not be the creators of humanity (or origins of higher consciousness). These groups believe in the existence of the supernatural and there is a much stronger focus on occultism and ritual than among the symbolic Satanic groups. These groups are still left-hand path so the intent is not so much to merge the individual will with that of Satan’s, but to model one’s own personal growth after that of Satan’s or Enki or Set or the demons. Worship is often on the basis of respect and emulation, rather than groveling or praying. Traditional or theistic Satanic groups include the Joy of Satan (and its offshoots), the Temple of Set (which no longer views itself as Satanic although it originated from a split from the Church of Satan), the Church of Azazel of Diane Vera, the Reformed Church of Satan (their website indicates that they are theistic but there are questions whether this group is still active),and demonolatry groups such as OFS.
Modern Satanism - This is wide category which is difficult to categorize. These groups and individuals often start with the concepts of LaVey, but have added other influences or have altered the practice. For example, they also tend to view Satan as a symbol or metaphor; however, often they are less likely to use ritual, if at all, and may have a much stronger focus on the.philosophical basis for Satanism rather than the “magical” aspects. For some, there is more effort to find more current and relevant symbology and techniques for 21st Century practitioners. Other groups may have a looser structure or nonexistent hierarchy. Many Modern Satanist organizations try to act as “Satanic Unity” organizations and try to create an environment welcoming to both “LaVeyan”/symbolic Satanists as well as traditional or theistic Satanists. Modern Satanist groups include the Modern Church of Satan, the Order of Cerberus, the Society of Shadows, the Sinagogue of Satan, and the now primarily defunct First Church of Satan and its split-off the Society of the Onyx Star (which remains active outside the U.S.).
Modern Luciferianism - These are individuals and groups that tend to focus on a symbolic Lucifer as a symbol of the self and as the bringer of light and wisdom which is seen as a more positive aspect than the darker, more carnal image of Satan. Unlike Satanism where there tend to be more symbolic or modern Satanists than traditional (despite the appearances sometimes on the Internet), Modern Luciferians are definitively the minority within the Luciferian community as most are theistic. Also, many modern Satanists also focus on the symbolic Lucifer archetype which means a good number of “Modern Luciferians” are found within the Modern Satanic organizations.
Gnostic Satanism or Traditional/theistic Luciferianism - These are groups that are based on Ophite Gnosticism. An early example is the Our Lady of Endor Coven of the Ophitic Cultus Satanas. This group was founded in 1948 by Herbert Sloane in Toledo, Ohio. This group predated the Wiccan trend in this country so it had more “Catholic” influence with modified ceremonies and prayers more consistent with its theology. Most modern traditional Luciferian organizations combine subsequent Western occult tradition gleaned from Wicca and ceremonial magic practices (a la Golden Dawn). Similar to traditional Satanism there is a strong emphasis on occultism and ritual workings. Some of these groups include the now defunct “Church Lucifer” and Children of the Black Rose founded by Nate Leved, and the Order of Phosphorus (which promotes the Luciferian witchcraft system developed by Michael Ford). The Church of Lucifer (which is different from Nate Leved’s “Church Lucifer”) as well as the Ordo Luciferi seem to be “Luciferian unity” organizations that welcome both modern and traditional Luciferians.