The article missed the most obvious Christmas reference in Shakespeare. Which explains why Christmas Eve is the traditional time for telling ghost stories.
Some say that ever 'gainst that season comes
Wherein our Saviour's birth is celebrated,
The bird of dawning singeth all night long:
And then, they say, no spirit dares stir abroad;
The nights are wholesome; then no planets strike,
No fairy takes, nor witch hath power to charm,
So hallow'd and so gracious is the time.
Christmas Day is a holy day for church and such and then the festivities begin after that. The Christmas season at court (with plays) begins on St Stephen's Day and lasts till Twelfth Night. (But all the way to Shrove Tuesday, on and off, for extravagant James.)
January 1st is when Lizzie gets her New-Year's gifts. But Lady Day (March 25) is when the calendar year starts for Elizabethans, hence all the confusing "1596-7" dates for court performances in some books.
When the English calendar shifts to line up with the Continent, the new year start date is moved to 6th April so the taxman doesn't lose any money for that financial year. The UK still sticks to that.