Simply put, traditional Satanism reveres Satan as lord, and God exists.
LaVey Satanism states that there are no gods other than yourself, because you are the creator of your own purpose and destiny.
The name was chosen purely for shock value.
Good question - one surely Prue-supposes the other. Although how someone can believe in satanism and not believe in the entity is a puzzle.
And the other puzzle is why is it that a person who believes in Satan and consequently Prue-supposes the existence of God then chooses Satan instead of God. True atheists of course don't believe in either.
Because theistic Satanists are a tiny minority of all Satanists. The vast majority today are LaVeyan Satanists. LaVeyan Satanism is explicitly atheistic.
And also because, in the long history of paranoid Christian claims of Satanism, and their persecution of people who were accused of it, no authentic examples have ever been produced. Even in the article you linked, there are statements about what such Satanists believe, but none are quoted as doing so.
Even the groups named are not rightly called theistic Satanists, but LaVeyans (atheists) or Noe-pagans. Some of them espouse the "left-hand path," but that isn't the same thing.