I find it hard to believe that Obi-wan would willingly train someone who was called ‘Darth Vader’ to be a Jedi. It’s one of the most villainous sounding names ever devised.
The problem with Obi-wan’s speech is that it’s simple enough without ESB and ROTJ:
Obi-wan had at least two students, one of which was later known as Darth Vader, betraying and murdering Anakin. Simple, effective, and good motivation for Luke to become a Jedi.
But with the rest of the story it becomes:
Obi-wan had a student, Anakin, who was seduced by the Dark Side into becoming Darth Vader, after which he hunted down and killed the Jedi. However, Anakin was not only seduced but betrayed and murdered by this aspect of himself called Darth Vader, implying an almost schizophrenic relationship with himself.
The problem with this more complicated version of events is that we’re made to think that Luke is very much like his father. This works with the first version of the story, where Anakin can be made to be a good man through and through, but not with this second story, where Anakin is betrayed by an aspect of himself which has been ‘seduced’ to the Dark Side. Among the dozens of fan theories I have read on this transformation, none of them rings particularly true to me, and I think that’s because Anakin can’t be both the good if impatient man that Luke is in ESB, and still be seduced by the Dark Side.
Anakin is the Schrodinger’s Cat of characters, at once good and noble, and also power hungry and evil.