Chistmas starts at Holy Eve and ends on Epiphany, January 6th, so New Year’s Eve fits into that time frame.
Saint Nicolaus’ December 6th according to the Julian calendar is probably on December 19th according to Gregorian calendar, thus, this is a different holiday.
Santa Claus is a straight copy of Sinterklaas. Even Wikipedia says so. Also, I don’t know any Dutch kids that believe in Santa Claus but a great many believe in Sinterklaas. Santa is not a gift giver here at all. If there are gifts at Christmas (a minority) then it’s given by family, not by Santa.
As a non christian, I’m talking about ‘the winter holiday where gifts are given’ which happens to be on slightly different dates depending on local traditions.
Basing the definition of what counts as ‘christmas’ on the church calendar isn’t very helpful in this global context where most people celebrating aren’t christian believers to begin with. And of course the christians borrowed from older traditions, which where also on different dates originally
