From what I read of Thursatru, it's not Satanism. It was
compared to Satanism.
Ahrimanism, I would wager to say, is the same;
comparable to Satanism, but the "anti-Zoroastrianism" as Thursatru is "anti-Ásatrú".
I would think "Left Hand Path" would be the most unspecific. As mentioned, "Satanism" invokes Satan, and thus binds to Abrahamism.
I see your point. And I also call my religion LHP - as I said I mostly use the name Satanism because the majority of texts about the LHP use Satanic terminology and because it's what the religion is best known as. It has become common use, I can't change it.
And most of the members of other "satanic" religions I listed would call themselves Satanists, or at least don't deny that they are. I know some that don't like the name, e.g. some Ahrimanists who don't like being called Satanists lest they get confused for atheists/LaVeyans. They incorporate a lot of Tantrism, though, as they consider Ahriman and the Druj a demonized form of Hindu deities. Another Ahrimani I know however calls herself a Satanist, some kind of henotheist and calls her deity Satan-Ahriman as she sees Ahriman as the historical predecessor to Satan.
Regarding Thursatrú, I have only read texts by one Thursian so far, namely by the author of Gullveigarbók (including that book). And while his main focus is on Norse mythology, he also worships Pan and considers both Pan and the Thursian deities manifestations of that same extra-worldly chaos he venerates of which also Satan is, according to him, a manifestation. Also, Tursatrú in the form I know it has developed from the writings of the Misanthropic Luciferian Order, a group of Chaos-Gnostic Satanists in Sweden who base their spiritualities on a variety of different myths (mostly Jewish, Sumerian and Norse, as far as I'm aware).
So you see, Satanism is a very diverse religion, and therefore, just because two forms of it look nothing like each other to you that doesn't mean that I wouldn't consider them both to be forms of Satanism.
Though in my experience it always results in an uneven comparison; a changing of the deity named to fit with the "dark deity" archetype. If one is just using a name - say, Loki, for instance - but what they're claiming to be Loki is nothing like the Norse God, can it really be claimed to be Loki? Can it really be claimed to even be Satan?
Good question. My own opinion: Myths are myths. They may contain truth, but they are not literally true, and not all are even metaphorically true. They can be used as a rather helpful tool to better relate to spiritual aspects of reality, and the forces/deities they describe might exist in one form or another. But we can hardly know for sure. So instead of searching which myths to believe I rather start with what model of reality to believe, and from that point I then look which myths, and which interpretations of them, help me best with that. Or I look which myths other people work with and see if I can relate to them as well and in which form they are compatible with my worldview and may give me more insights on it or inspiration to modify it. It's spirituality, not reconstructionalism after all.
How is "Promethianism" Satanism? Wouldn't that just be worship of Prometheus? I worship Prometheus and Zeus, and so did the Ancient Greeks.
Okay, with Promethianism I leaned a bit too much out the window. I don't know much about it actually, it seems to be a different name some Luciferians use for their religion (and don't ask me whether that is the same as Satanism or not xD).