Julian refers to Julius Caesar, who declared a new calendar when he became emperor. But the Julian calendar accounted for 365.25 days per year. In 1582, pope Gregory XIII corrected the calendar to account for 365.2425 days, which is closer to the actual value of 365.2422 days. His calendar is called the Gregorian calendar.
The Gregorian calendar moved christmas forward 10 days, which made the old date of celebration Jan 5. In 1800, it moved to Jan 6, the Feast of the Epiphany. The 12 days of christmas are the 12 days between Dec 25 and Jan 6, inclusive.
The Julian christmas is also known as little christmas or old christmas.