You’re right, though the Sith are a group of Dark Jedi, so from an outsider’s perspective they’re effectively the same. Vader is also a former Jedi himself anyway.
It should be stated that all Sith/Dark Jedi were former Jedi.
I don’t think that’s necessarily true. The Emperor wasn’t.
Your right. That would mean he didn’t have to be a Jedi to start off as a force user.
Also, for a better explanation for why Vader and The Emperor were the only Sith/Dark Jedi during the OT is because the Jedi defeated all the others with Palpatine’s rise to power being because of his strength with the Force, his ability to see the future, etc.