I hadnt heard about the Christmas witch part.
These people are nuts!
La Befana ( usually just called Befana ) is the Italian version of Santa Claus. Yes, she is a witch, but she is the one who gives presents to good children and sees who is naughty or nice. This used to take place at Epiphany and her name is a corruption of her original name: EPIPHANIA.
An old woman, Frau Berchta, was the one to bring gifts to good children, in Germany and Austria.
America has Santa Claus because the Dutch, who first settled New York, was the inspiration for the famous poem THE NIGHT BEFORE CHRISTMAS and this scharactesr bears only a slight resemblance to our present day Santa.
The English had/have Father Christmas, who is skinny, BTW, so American descendants of English and Dutch settlers were comfortable with this figure. And subsequent immigrants adopted the American traditions, that they found here.