Obi-Wan left Anakin for dead, and one movie later (in A New Hope) he was saying Anakin had been killed by Darth Vader.
What exactly did Obi-Wan know about Anakin and Darth Vader before A New Hope started?
Vader tells him.
In Obi-Wan Kenobi - which takes place ten years after Revenge of the Sith - Obi-Wan learns about Anakin/Vader's survival from Reva, one of the Inquisitors. In the final episode, he faces off against Vader, incapacitates him, and attempts to pull him back to the light side, only to be rejected thusly:
VADER: Anakin is gone. I am what remains.
OBI-WAN: I'm sorry. I'm sorry, Anakin. For all of it.
VADER: I am not your failure, Obi-Wan. You didn't kill Anakin Skywalker. I did.
OBI-WAN: Then my friend is truly dead.
Vader believes that he has "killed" Anakin and taken over his body, and Obi-Wan believes it too, hence him telling Luke that Darth Vader "murdered" Anakin.
During the first days of his exile, Kenobi visited the Lars Homestead every day, always trying to stay clear of Owen and Beru so that they would not see him and watching Luke from afar. It was during these visits that he would contemplate on Skywalker's "death", though he still had an inkling that he was in fact alive as well as wondering why his former Master, Jinn, had not yet contacted him from beyond death.
These doubts were then later verified:
For the first few months of his self-imposed exile Kenobi had no idea what had become of Vader after he had left him to his fate on Mustafar, but one day at a cantina Kenobi finally heard that he was alive when news reached him of Vader’s victory at the Battle of Kashyyyk. He was shocked that the remaining Jedi thought they could band together to defeat the Sith Lord. He had hoped that they would go into hiding like he and Yoda did. (source)
So why did Obi-Wan say that Anakin was killed by Vader? Well, metaphorically Anakin died and was reborn as Vader, but I doubt Obi-wan was saying Vader killed him just to wax poetic. Instead of telling Luke the exact truth, Obi-Wan told him what he needed to hear to help guide Luke on the path to become a Jedi. Telling Luke that his father was a great Jedi knight who died in battle fighting against the Empire was just the sort of thing that could help drive Luke to leave Tatooine and ultimately join the Resistance.
Imagine if he had told Luke exactly what happened...
Hey, kid. Your dad's still alive. Oh, and by the way, he's the scourge of the galaxy. Also, he killed your mom.
That might not have played out very well. It wouldn't have been particularly motivating for a young and impressionable Luke to learn such a dark and hard-to-accept truth. He might have given up all thoughts of joining the Academy, or later the Resistance, knowing he might one day have to oppose his father.
It isn't explicitly mentioned in the films. But I think the implication is there that Obi-Wan, Yoda, Bail Organa, et. al. all knew the full truth about what happened.
In Star Wars: A New Hope, Obi-Wan's lie about Anakin being "killed" by Vader certainly implies that he explicitly knows that Anakin became Darth Vader. If, at that stage in his life, he'd believed that him leaving Anakin for dead had killed him, he'd have said that.
In other words, at some point between him leaving Anakin half dead and Luke finding him on Tatooine, he (and presumably Yoda and his other allies) discovered Anakin's fate.
Obi-Wan could had administered the coup d'grace, and probably kicked himself repeatedly over the years for not having the guts to do so. Obi-Wan was both exhausted and heartsick over the slaughter of the Jedi order and younglings, and realizing that his own apprentice was responsible. But even after seeing the Jedi Temple holos showing what Vader did, and seeing the same man nearly kill Padme (which not only for profound admiration and friendship, but also perhaps in his unrequited love for Siri Tachi and/or Duchess Satine, may have been carrying a torch for as well), Obi-Wan couldn't bring himself to dispatch the young man he'd considered as a brother. Obi-Wan, in his shock and grief, left his apprentice to die of his wounds and burns, and he also needed to tend to the stricken Padme. He didn't realize that Palpatine would come to Vader's rescue, and Vader's own superhuman endurance to not succumb to the injuries that would have finished off virtually anyone else. Even a powerful Jedi like Obi-Wan is "human" and has faults. If he employed sophistry, and tried to palm off his deception of Vader's status as father to Luke with that "point of view" crap, at least understand that his solitude on Tatootine, with nothing but that awful memory to deal with, probably drove Obi-Wan to the brink of insanity. Little wonder that he knowingly went to his own demise on the Death Star willingly; it was relief from his pain.