I have to correct you on the Italian side. We have Santa!

Let me explain better. Depending on the region in which you are you have one of this four traditions:
1. Santa that comes the 24th in the night with presents (there are many Santas that can be booked to go in the houses with gifts for the childrens)
2. Befana as you said
3. Jesus child that comes the 24th in the night with the presents
4. The three kings that comes with the presents for the childrens, in a sort of Biblic revival