Apparently, this is something that’s going on on twitter? I’ve seen references to it, but I haven’t read any of the discussion myself, so if I hit on something or am off on a completely separate track, I have no way of knowing, this is just based on my own discussions of this topic!
I think there’s value in the words “Anakin and Vader are separate” in the sense that it’s shorthand (admittedly very clunky wording for shorthand, but this is how I tend to see it) for how Anakin and Vader are very separate mindsets. That what you can expect from one, you cannot expect from the other. I can see the value in stating it along these lines, because I think (for example, when Dave talked about this) crew and creators have to sidestep confirming anything too specific about mental states, like I don’t think they want to state that Anakin and Vader are literally separate personas, because they’re not trying to state that Anakin has DID.
It’s a lot harder to explain that of course Anakin and Vader are the same person, that Vader’s issues are literally Anakin’s issues taken to a new extreme, but that how they think and feel are significantly different, because the dark side warps your mindset the more you seize onto it, and Vader has absolutely soaked the dark side into himself.
But they’re still the same person. Vader’s refusal to look at the responsibility for his own actions–not even his actions pre-fall to the dark side, but just as Vader, when he helped enact the genocide against the Jedi, when he murdered the culture that adopted him, when he murdered their children–as an inability to face what he’s done and how he knows it’s wrong, that’s straight from Anakin Skywalker.
George Lucas’ directions to Hayden wouldn’t make sense if they were literally separate people:
If they’re literally separate people, then Darth Vader is brand new and literally just born in Palpatine’s office, and why would he care about Padme?
Darth Vader doesn’t start when the mask comes down (though, even if it had, his first action is still to ask about Padme and then wreck the room in a fury of a temper when Sidious says she’s dead), he starts when he kneels to Palpatine and says he’ll do whatever he asks, just help him save Padme’s life. That’s Darth Vader right there.
^ THAT’S DARTH VADER RIGHT THERE.
It’s “Darth Vader” who marches on the Jedi Temple and kills the children. It’s “Darth Vader” who murders the Separatists leaders. It’s “Darth Vader” who fights Obi-Wan on Mustafar. It’s “Darth Vader” who cries on Mustafar as well, because he knows this is wrong. It’s “Darth Vader” who rushes to see Padme but then chokes her when she backs away from him. It’s “Darth Vader” who screams, “I HATE YOU!” at Obi-Wan because he loved Obi-Wan so much.
If they were separate people, even separate personalities, Vader wouldn’t be doing any of these things or having these reactions.
It’s the same in the comics–literally every comic that has Darth Vader as a central character for any significant amount of time has him talking to Obi-Wan’s memory, has him trying to resurrect Padme, has him having bitchy revenge fantasies about dunking Obi-Wan in lava instead, has him screaming, “If you really loved me, you’d have killed me!” at Obi-Wan’s memory, has him obsessing over Luke specifically because he’s Anakin Skywalker’s son, has him losing his shit because someone looks very much like Padme, has him pausing because he knows Ahsoka Tano.
Darth Vader is Anakin Skywalker every step of the way. The dark side has twisted him so that he’s unreachable by these people–but so much of that is precisely because they knew him as Anakin Skywalker and he cannot stand them knowing who he was (is) and what he did, that the reason Padme can’t stop him, the reason Obi-Wan can’t stop him, the reason Ahsoka can’t stop him, is because Vader knows they know exactly who he is. He’s so afraid of facing himself that he can’t tolerate them precisely because he’s very aware of who he is in a way that he doesn’t want to acknowledge.
He tries very hard to smother that part of himself, but if he were at all successful, he wouldn’t be so damn afraid of it. And Darth Vader is terrified of that part of himself.