Ender — What you wrote is mainly true. But my humble understanding is that the Pagan Holiday was December 21st or 22nd, the day of the Winter Solstice.
I heard from some shiurim that the early founders of that faith decided to synchronize the date with our Chanukah (25th of Kislev) since they understood the miraculous nature of 25 (for us).
Notice that December 25 through January 1 (inclusive) is 8 days long. Perhaps that the initial reasoning why the compiler(s) of the present civil calendar made sure December has 31 days.