Sith are different from Dark Jedi, yes. for IRL analogs, the Jedi Order is like the catholic church around Luther's time. The Sith are satanists, Dark Jedi are like Protestants.
Dark Jedi still perceive the Force under the paradigm of the Jedi, they just choose to embrace the Dark Side.
the third type of Jedi- Gray Jedi probably existed inside the Order itself- people like Jinn and Revan.
I had always thought it was an "all Sith are Dark Jedi, but not all Dark Jedi are Sith" thing. In that Dark Jedi are any force users that embrace the Dark Side, while the Sith represent a specific lineage/tradition of Dark Jedi, tracing back to the Sith species and the Sith Empire.