The following thoughts are asked from a fully assimilated, 'saga' perspective. I appreciate the convoluted history of the story, the improvised nature of which largely causes these questions. But I am thinking about these things in light of all material up to the present.
The prequels and Clone Wars series establishes Anakin Skywalker as a relatively well-known war hero. The novelization of Episode III makes it sound like Anakin and Obi-Wan are two of the most famous Jedi around. Yoda is seemingly the prime member of the Jedi Council. The holonet seems to function like the internet does for us.
The Empire suppressed information and spread propaganda, but a whole galaxy can't just forget what was so well known after 19 years. Vader's identity is not public knowledge, and I believe the official story is that Anakin was killed defending the Jedi temple.
Fast forward. Luke Skywalker is told that his father was a navigator on a spice freighter. Obi-Wan seems to bring Luke up to speed by giving him more or less the 'official' version of the story. Anakin was famous, so I wonder if Luke ever heard of him elsewhere. If so, would he have had the realization offscreen that his father was THAT Anakin Skywalker? If not, how did that information become to thoroughly suppressed? Obi-Wan was willing to let Luke know the seeming 'official' version of Anakin's story, but Owen dreamt up something much more mundane.
The deleted scene with Red Leader, Biggs, and Luke that was partially reinstated for the SE references Luke's father. It seems like others in the galaxy know who Luke's father is by merely knowing his surname and Jedi status. Is the lack of knowledge that his father was a Jedi sufficient reason for Luke to be taken in by Owen's lies? Again, does this mean that he must have realized later, "Golly, my father was THE Anakin Skywalker?!"
Related to that, why does Luke not know what Yoda looks like? Is it because he simply did not have time to research it between hearing about him and leaving for Dagobah? Surely Luke would have done a galactic Google search about the Jedi and have seen (even propaganda-spun) information about some of the most famous Jedi.
I'd like to see these issues addressed in the upcoming Heir to the Jedi novel. I've always found it confusing about who in Star Wars knows what. This issue was really driven home in the pre-prequel EU, because the prequel story had not been told yet so there was nothing for authors to have the characters know. Since the post-ROTJ EU was written in broadly chronological order, the time at which the prequel films were released coincided in-universe with discoveries of old information that had been lost. Now that the EU has been "rebooted," as it were, I'd like to see novels tackle this issue more directly.