Kirk has just called her "Mrs. Sarek," i.e. assuming that "Sarek" is her married surname. It therefore logically follows that the "Vulcan name" she's referring to is her true married surname, i.e. Sarek and Spock's family name.
Assuming there's such a thing as a surname in Vulcan tradition. "Vulcan name" could just as well be what she was told to adopt instead of Amanda, or instead of Grayson, or instead of both - the classic "Indian name" thing.
As for whether DC Fontana thought or didn't think this or that, why would that be a concern here?