Depends on how you interpret the Golden Age text. But you're correct that the former is the light bringer and the latter is adversary/legal opponent.
But the silver age crap messed with that canon.
My head canon is that Lucifer was a member of the Q Continuum while Satan was a misunderstood Time Lord.
The Pope needs to have a sit-down with the folks at Disney to learn how to properly manage a shared imaginary universe.