Happy Birthday to her, she got the best present in the whole world - a dad who cares beyond the world for her.
Good on you for going around even though she was coached - as
@winger said, she may not remember what she said when she looks back but she will remember Dad was always around on my Birthday.
I'm sorry you had to see your daughter being coached at 4 to talk like that, it's heart breaking to see such a young kid being put in that frame of mind.
You cope by sharing and caring, put that love into those little Christmas presents, date them with an object from this Chrustmas like a newspaper clipping - then box it & save it for future her.
There are plenty of children out there with no parents, maybe volunteer this Christmas to support them, if that's not too emotional, will be your way of sharing all your love as a gift from your daughter to them
We're all here for you buddy
MJ