Vader in the OT never references Anakin in any direct way. He only references him in the ESB SE during the new Palpatine segment

He only says "That name no longer has any meaning for me". That part of himself is gone, or so he thought. But that's not really referring to himself in 3rd person.

The Rebels Season 2 premiere has him referring to Anakin in the 3rd person. I think that's the first time we hear Vader refer to him as a separate person. I'd say that's a rare error on Filoni's end.

As for the republic knowing about Anakin, that character basically disappeared after Order 66. It's doubtful most people of the galaxy put Vader and Anakin together as the same person.

As for Luke Skywalker, maybe it's a common name or not a rare name in the galaxy. Or the people on Tatooine like Wormie and Fixer just don't care. When ANH was written, Luke's father was simply a great pilot and jedi so his son living on some backwater planet is not that far fetched.