NeverarGreat said:

Nobody seems to recognize the name Skywalker in the OT. Even Leia, who knows the history of Obi-wan Kenobi, doesn't seem to understand the significance of Luke's name. Lando is totally ignorant of it as well, and he may have been around 20 years old at the time of Anakin's fall (though I think that the time of his fall should be pushed back at least ten to fifteen years in this rewrite). This seems to suggest that Skywalker was not a well-known name in the galaxy, or at least people didn't attach significance to it. I'm guessing that Anakin Skywalker is the name that his parents gave him, because I doubt that Anakin would be a name chosen by him as a Jedi title. Remember also that according to ANH, Anakin was not a Jedi when he left Obi-wan, so Obi-wan could not have given him any Jedi name.

Given that the Jedi themselves have slipped into myth, I think it would be safe to assume that the names of the Jedi themselves would also have fallen out of general knowledge. Not to mention Vader himself would have wanted the name Anakin Skywalker to be eliminated from all knowledge. And I'm not sure what you meant by Anakin not having been a Jedi when he left Obi-Wan. Even though Darth Vader says "When I left you I was but the learner." I don't think this negates the idea of him having a Jedi name. Even if he wasn't considered a full Jedi at the time, he would still have been a member of the order. Also, in ANH Obi-Wan, when telling Luke how his father died says, "A young Jedi named Darth Vader." So, when Vader says "I was but the learner." I think he was just referring to him being less skilled than Obi-Wan at the time, but that now he was stronger.