I think that someone asks OB-1 a favor in EpIII, but he refuses because he is still down from the failing of Anakin. So someone asks his help
"Help me Obi-Wan Kenobi, you're my only hope."
"Mylady, you're mistaken, I'm Ben Kenobi, my brother died years ago!"
Anakin doesn't fall in II, in fact he may not fall until very late in III, at least completely where everyone knows he has turned.
Rehashing an old line from ANH, Eugh....
As for the Ben Kenobi name, he uses that on Tatooine, not to hide from his friends but to maintain a low profile.
Episode-III takes place 2-3 years after Episode-II, so a line like "He dies years ago" would be out of place.
Anakin ends Episode-II still pretty much on the side of good, with hints of his emerging evilness throughout the movie. Though he may be a few lbs lighter near then end than he was at the beginning.
YIt's not like he stands in the center of Mos Espa shouting "Who-hooooooo, Obi-Wan is in da house!"
And he hardly goes to Tatooine at all until he settles there.
In Episode-I he never left the ship, he doesn't go to Mos Espa or any where else for that matter.
He doesn't visit Tatooine in Episode-II.
He probably does in Episode-III, but only to deliver Luke to the Lars and possibly to begin his hide out.
The only knowledge of Obi-Wan that the denizens of Tatooine have is over the news, if they ever talked about the great Hero Obi-Wan Kenobi.
Remember, he hides his house far away from anyone esle, on the edges of the Jundland wastes, where people hardly ever visit. I'm sure that any unwanted visitors could just be "convinced" they had never been there.
He is referred to by Owen as a hermit, I would imagine that he spends nearly 20 years on Tatooine with very little contact with anyone.
If the Empire does hunt down the Jedi then I would think that there would be quite a few greedy people willing to turn a jedi in for profit..and remember..Luke didn't know that Ben was a jedi. So it goes along that feeling that Obi-Wan changed his name for safety reasons. Plus the Jedi come to be ridiculed by the common people..maybe saying you're some big time Jedi Master is just asking for someone to try something.