You are correct. However, back then, they used what is called the Julian calendar, which doesn’t account for leap years. The winter solstice on that calendar back then fell on Dec 25.

https://en.m.wikipedia.org/wiki/Julian_calendar

This is because Jan 6 was the original date for Christmas. The Armenian Church still observes Christmas on Jan 6.

Why did Christmas move to Dec 25?

> The main reason, however, seems to be as follows: the 25th of December, in the Roman Empire, was a great feast date connected with the “Birth of the Sun” known “Saturnalia,” since just after the Winter Solstice, the days having reached their shortest limit, begin to grow longer and longer. Because of the promise on that day (25th of December) of longer days to follow, there were great rejoicings. So the Church thought to prevent the faithful from attending the “abominable” celebrations of this very popular pagan festival on that particular date.

Source: http://www.stthomasarmenianchurch.com/religious-information/saints-feast-days/why-we-observe-christmas-on-january-6th/