When Anakin fell to the dark side and took up the title of Darth Vader, he still acknowledged his identity as Anakin Skywalker: the name represents his identity as a person in whole, not just of the light (not yet anyway).
At least, we know that to be true up until Mustafar, since he didn't say anything to the effect of "that name means nothing to me now" to Padme or Obi-Wan.
At some point after that, he decided that Anakin Skywalker represented the light in him, and chose to "kill" him and deny Anakin as part of his identity, assuming Vader as his full and sole identity as the darkness within.
When did this change happen, why did it happen, and why did it happen when it did?