That was obviously the intention in ANH, but only because they hadn’t come up with the “Anakin became Vader” twist yet: “Darth Vader” was supposed to be his actual name. But once the father twist was written in ESB, Obi-Wan suddenly started saying “Vader” instead of “Darth”, implying that “Darth” was now a title given to Vader once he turned to the dark side.
If Darth Vader was meant to be his actual name then after they came up with the “Anakin became Vader” twist, it should be implied that this is the new name that he took on when he turned to the dark side. In ROTJ when Luke says “I’ve accepted the truth that you were once Anakin Skywalker, my father” Vader says to him “That name no longer has any meaning for me”.