When George Lucas made Star Wars, he may have had some tentative ideas for sequels and prequels, but as those started taking shape, there were inevitable contradictions. That was definitely the case after Revenge of the Sith because while it's easy enough to see how that leads into A New Hope, there are certain things Lucas couldn't - or didn't bother - help make sense.
Well, today's Obi-Wan Kenobi finale goes some way in making up for that, and not in a way that feels forced or overly convoluted (a relief after some of what we saw in Disney's Star Wars sequels).
Why did Ben Kenobi "lie" and tell Luke Skywalker that Darth Vader murdered his father? Well, after the Jedi Master's exchange with the Sith Lord at the end of their battle, it now adds up. In what appears to be a brief moment of clarity, Vader frees Obi-Wan from taking any blame for his descent into evil by making it clear that Darth Vader is the one who "killed" Anakin Skywalker.
Obi-Wan calling his former Padawan "Darth" on the Death Star is also addressed in a subtle way when, realising Anakin is well and truly gone, he walks away from Vader by saying, "Goodbye, Darth." It feels like an insult in the sense that it shows just how far the former Jedi has fallen.
When it comes to why Vader's rage appears towards his former Master appears to have subsided when they meet again in A New Hope, that can be explained by his exchange with Emperor Palpatine. The villain insinuates that Vader's feelings for Obi-Wan have weakened him, and knowing that makes him vulnerable on multiple fronts, he appears to forego his mission to capture and destroy the Jedi Master he once called a friend. With that, we're assuming he focuses on increasing his own power, hence why he considers himself the Master during their next confrontation.
There are other obvious connections like why Princess Leia chooses to ask Obi-Wan for help, as well as how Vader's quest for vengeance in some ways leads to the Rebellion's creation (they escape when the Sith Lord decides instead to pursue Kenobi).
However, one fun theory is that the large scar on Anakin's forehead when he unmasks in Return of the Jedi was, in fact, caused by Ben in this battle. We're not sure what to make of that, but it's not outside the realm of possibility and would mean Obi-Wan left his former pupil with another scar to remember him by.