Load of cobblers, complete garbage, I'm afraid.
The "original" Christmas date is unknown. The 6th January was, and still is, the feast of the Epiphany when the wise men came to visit Christ. The Church in Rome instituted Christmas to counter those who claimed that Christ was not really God made Man but only became Divine when he was older.
There is NO evidence that December 25th was any sort of pagan festival before Christians chose it for Christmas. You can find images of ancient Roman calendars from around the time of Christ. These show no festival on Dec 25th. There is NO record of any birth day for Mithras. Even the supposed "Natalis Invicti" feast day is dubious and, if it existed at all, came after Christmas started, Romans did not celebrate the Solstice - their festival around that time (Dec 17th), Saturnalia, was a harvest festival.
The Feast of the Resurrection, Easter in English, was celebrated by the church from its earliest days. There are records of Easter services from the 2nd century. There was a debate in the church about which day to celebrate which settled on the Sunday after Passover - Sunday being already know as "The Lord's Day". Because of its link to Passover, the feast was named the Christian Passover - Pascha in Latin (in modern times Pascua in Spanish, Paques in French and so on).
Over 400 years later, Christianity reached the Anglo-Saxons, they were taught about the Paschal Season which happened to coincide with their month of Eosturmonath. The ordinary people therefore translated Pascha as Eostur, later Easter, though the church, which used Latin, continued to call it Pascha until the reformation.
The early English writer Bede speculated that Eosturmonath had, many years earlier, been named after a goddess Eostre whose worship had died out. Unfortunately, absolutely nothing else is known about this supposed Goddess. She appears in NO anglo-saxon myths or legends, she has NO known feast day, she has NO known temple or site, NO known dedication.. Modern scholars have concluded that she was probably made up by Bede to explain the month name and was never actually a goddess at all.