We don't really know what day Christ was born. There are some distortions caused by the bias of the historian, yes.
All historical writing is subjective.
But in the bit I have quoted above the distortion goes beyond this, by distorting easily ascertainable facts. The Gregorian and Julian calendars differ by 13 days. The Gregorian calendar was first introduced in 1582 by the Roman Catholic Church, and Protestants and Orthodox did not accept it until much later. The Russian Orthodox Church still follows the Julian calendar, so in Russia Christmas will be celebrated on 7 January. These are easily ascertainable facts and could be checked in most almanacs and encyclopaedias.