Yes, you've got it.

The ancient Church moved Christmas to December 25 pretty early on, somewhere in the fourth or fifth century I think. All Churches and demoninations celebrate on December 25 today. The difference is only in which calendar they use.

When the Gregorian Calendar was adopted by Rome in 1582 it was in force only within the Roman Empire. Setting the calendar right in 1582 meant removing 10 days from the month of October that year and fiddling with the leap year adjustment so the calendar would stay in synch with astronomical observation.

The Eastern Churh and, I believe, the Church of England—created in 1534—didn't immediately adopt the new “Papist” calendar. Not only was it Roman Catholic, it meant that important dates in “God's sacred calendar” were now gone. Over time, governments adopted the Gregorian calendar for practical reasons having to do with synchronicity and commerce but many churches never adopted it. The last government to accept the Gregorian calendar did so early in the 20th century.

Since Christmas is a religious holiday, it's date was set by the Church, not by governments. When governments began adopting the Gregorian calendar, Christmas started showing up in January, at least until the Church adopted the Gregorian Calendar as well.

      - Mark