Choosing year 1 for the AD/BC calendar, new years day, and the date for Christmas are different things, and only poorly related.
In the Roman calendar as of Julius Caesar's time, the solstices and equinoxes fell on the 25th - Dec 25 shortest, June 25 longest etc. But an error in leap year counting produced a shift of about one day per century. By 4th C AD, the shortest day was Dec 21. By the time of Pope Gregory in the 15th c the error was 12 days. The Gregorian calendar eliminated that error, and corrected the leap year counting. So now we are stuck with Dec 21 as the approximate winter solstice.
As told in Eastern Orthodox churches, Theophany, Jan 6 was the 2nd most important church feast day, and it celebrated all the 'beginning' events (Pascha/Easter is the most important). Then following the example of Rome, the birth feast was split off, and celebrated 12 days earlier, on Dec 25, the traditional winter solstice. In the east Theophany remains important, but in the Roman church it just celebrates the Wise Men.
There are several church feast days defined by choosing Dec 25 as the birth date. 8 days later is his circumcision, Jan 1. March 25 was his conception (or Annunciation to Mary in Luke), Jun 25 is John the Baptist's birthday (6 mths older). These feast days were set sometime around 400ad.
The AD/BC counting was introduced in the 6th c. The claim that it is off by 4 years is based on fact that, by modern counting, Herod died in 4 BC. His death is the only Gospels event that can be tied to extra-Biblical evidence. Luke mentions a Roman governor and a census, but that governorship was 10 years after Herod, and there's no external evidence of that census. One of the Gospels says Jesus was 30 when he started his preaching and a timeline constructed from a different one indicates it was 3 yrs long. That does, roughly, put his death during the rule of Pontius Pilate (26-36AD).
The New Years WIki article says that various dates were used as New Years. March 1 was common, and some of the month names reflect that (Sept, Oct, Nov, Dec, 7,8,9,10). But Jan 1 was also common in Roman usage, but it was never prominent in Church liturgical calendar. As I wrote earlier March 25, Lady's Day (Annunciation) was used as New Years in the English speaking world until just before the American Revolution. George Washington was born in a 'conception date' Julian year, and died in a 'circumcision date' Gregorian year.
discusses in detail how and why Dionysius came up with the AD counting in the 525AD. His main interest was in creating Easter date tables, not in setting Christmas or New Years. Choosing the date for Easter was controversial in the 4th c, and even now there's a difference between how Western (Roman) and Eastern churches set it.
The use of CE/BCE, common era, is just a way of saying that the year counting is based on common accepted usage, as opposed to the practice of one church's liturgical calendar. The numbers are the same.