Who's to say that the general public would have known that Anakin did anything. It was probably assumed by the general public that Anakin died with all the rest of the Jedi when they "betrayed" the republic. After killing Mace, Anakin pledges loyalty to Palpatine and he takes the 501st to the Jedi Temple to kill all the Jedi while Palpatine issues Order 66 to kill all the other Jedi. Anakin is also cloaked and the operation is carried out under the cover of darkness so it's likely that if anybody saw the Jedi heading toward the Temple, they likely would not have known who it was. The only recordings of Anakin's dark deeds were made at the Jedi Temple and possibly at Mustafar. But since the Jedi Temple was under martial control by the Clone Troopers, who by the way, told any outsiders that came by that there had been a rebellion, then it would be reasonable to assume that these recordings were covered up and any Mustafar security recordings that may have existed were probably either destroyed by the destruction of the facility or seized by the Clones when they arrived with Palpatine to rescue the burned Anakin. So really the only people who knew what Anakin did where the clones who carried out the sedge and rescue missions and they would not have disclosed classified military information. Yoda and Obi-Wan obviously knew and it's reasonable to assume that Bail Organa could have known of Anakins betrayal but as Obi-wan and Yoda had already parted ways by the time Vader would have been revealed then he would have had no way of knowing Vader was Anakin as he assumed that Anakin had been left for dead. And, even if Ob-wan, or Yoda, had specifically said, "hey, Anakin's been calling himself Vader these days" and Bail had put it all together after this dude Vader starts showing up then it still doesn't mean that he would have started spreading it around like wildfire enough to reach the ears of Luke Skywalker living on a backwater planet with little ties to the dealings of the Empire. The planet was outside the republic under the control of the Hutts until probably very recently before A New Hope. Remember the Luke and Biggs conversation about the empire won't even mess with this rock and things can change, you know they've already started to nationalize commerce, and pretty soon your uncle will be a slave working for the greater glory of the empire or something like that.