My read on it is that Obi-Wan told Owen & Beru that Anakin died with the other Jedi, without really elaborating on the details, and details were not sought. After all, Owen's connection to Anakin was mostly though his loyalty to the memory of Shmi. He barely knew the man he became.
Insofar what the public knew post-RotJ; I don't think an elaborate cover story was necessary. While the connection between Luke and Anakin, and by extension Leia would be self-evident; there's no obvious connection between Anakin and Vader.
So far as any public record might be concerned, Anakin Skywalker died at the end of the Clone Wars. That would have been the Empire's official story (though even then only in sealed security files; the public records of all Jedi would likely have been erased entirely), and there'd be no reason to doubt that account later down the line.
Vader on the other had was always a mystery, even to the higher echelons of the Empire. Even the ISB likely had no clue who he was, and knew better than to poke too deep into the Emperor's own secrets. Those reckless or ambitious supervisors that might have, likely ended up never being heard from again.
Who knows what, if anything Luke said to the Rebellion about his confrontation with Palpatine. All anyone besides Leia (and probably Han) knew; he was captured before the assault on the bunker, taken up to the Death Star, and escaped in the chaos of the attack. I mean it's not like it'd be the first time he'd ever escaped from a Death Star, so why would anyone question it?
