In my understanding, the mother pretended to be her daughter after 2 years passed, to easy their pain, (more like confusion) But gave it away on the marriage when she scratched her husband chin.
At first after this event I thought the mother managed to return to this very special event, but she stayed silent when he asked "it was you all this time?" so I concluded it was her all the time.....
Movies and books have may interpretations, is up to the person watching reading to do that. But I guess there's always the "official" writer/director interpretation
Edit:
Thanks Prince for hosting those movies and Gokusen on your site, that Korean movie, A Day, is a sugoi tear jerker, haven't cried that much since watching "A Scene at the Sea"