Obviously what happened was Lucas wasn't really sure what the deal was. Probably the main reason he deleted the mention of red leader referring to Luke's father in the SE because he didn't know what he would do with the Anakin in the PT. If you go by the old EU things are certainly complicated. Been reading a bit and apparently Anakin and Obi-Wan were like the biggest heroes of the clone wars. But there isn't actually any indication of that in the films. From the perspective of the movies Anakin could easily have been seen as just another jedi, and one who was killed by Vader during the purge. OT or PT Tatooine is a remote planet that likely doesn't get many jedi visitors so I doubt anyone there would know of Anakin.

As far as the split personality thing goes, as others have said in the movies it's more like "Anakin isn't my name anymore." Referring to himself in third person as he does in the EU (and the Emperor sort of makes it sounds like they're two different people in the SE) is more of a metaphorical thing that fits with Obi-Wan's certain point of view speech. As for why Vader seemed to adopt a new identity unlike Palpatine or Dooku, well those two kept their names for political purposes. Anakin could have still pretended to be Anakin after he turned (the propaganda being that the hero of the republic has become the hero of the empire) but with the addition of the suit I think he was given a rebranding as a symbol of terror that strikes fear into the hearts of the remaining jedi and rebels. If that's what Palpatine was going for with the suit makeover then it makes sense that the hero Anakin persona would be dropped in favor of a cold and faceless imperial killing machine.