Wookieepedia's article on Anakin cites the Legends novel Luke Skywalker and the Shadows of Mindor regarding the widely accepted story of Anakin's "death". This book was written by Matthew Stover (who also wrote the Revenge of the Sith novelization), so Stover's explanation is likely what Lucas had in mind. @Richard has helpfully provided a quote from the book:
“Kid, in the Clone Wars, everybody knew him. He was the greatest hero
in the galaxy. When he died, it was like the end of the universe.”
Nick’s gut twisted again at the memory. “It bloody well was the end of
the Republic.”
Luke stopped. He looked like something hurt. “When he … died?”
Nick came to a halt gratefully, bending over with hands on his knees
while he tried to catch his breath. “Way I heard it, he was the last
Jedi standing in the Temple Massacre—when Vader’s Five Hundred First
went in and killed all the Padawans.”
“What?”
“That’s where your father was killed: defending children in the Jedi
Temple. He was not only the best of the Jedi, he was the last. Nobody
ever told you the story?”
Luke’s eyes were closed against some inexpressible pain. “That’s … not
the way I heard it.”
This is may not be exactly the Empire's official story for Anakin's "death" since it portrays the stormtroopers in a bad light (killing mere Padawans). However, since this story is widely accepted (Nick is surprised that no one told it to Luke), it's likely that the Empire told everyone that Anakin was just one of many Jedi who died in the attack on the Jedi Temple by the 501st.