My mistake. Sometimes I wish they chose more distinct names

No they still get called that because Christians call lots of things Satanic. To Satanists that's part of the game. They're reveling in the pearl clutching that claiming to literally worship Satan causes. The actual focus of their worship is more a philosophical ideal of the self and spiritual independence.

The new Satanic Temple reveres a wider pantheon of pagan dieties that the Christians all called Satan but had their own individual cults. It's more of a safe place for them to practice their own faiths while in community with eachother.

Islam and Judaism believe in the same old testament so there's two other possibilities.