Seeing as one of the supposed aims of the SEs and a clear aim of the revisited saga is to try and make the PT and OT fit together maybe it would be possible to change the line where Obi-Wan says he had a pupil “named” Darth Vader and just have a pause so it goes :

BEN : Darth Vader, who was a pupil of mine until he turned to evil, helped the Empire hunt down and destroy the Jedi …

That way Vader is not necessarily his name but could be the title given to him as seen in the PT. We also don’t know if this is Anakin or another pupil he had before Anakin if his youth at that time isn’t mentioned.

If he is a pupil of Obi-wan it doesn’t need to be said that he is being trained as a Jedi. It’s not like Ben also teaches the oboe 😄