This isn’t a full answer, but some more pieces of the puzzle.
Briefly: the OED supports the argument that the usage “twelfth night” =
“Jan 6th” comes not from subtleties of ecclesiastical reckoning, but from a recent shift in meaning.
According to the OED, the twelve days originally referred to the twelve days after Christmas, i.e. starting from the first day after it; the somewhat misleading phrasing the twelve days of Christmas to refer to these days only comes later.
Hence the twelve days have always properly been Dec 26–Jan 6, the first ‘day of Christmas’ is not the same as Christmas day, and twelfth-day is a synonym for Epiphany.
(These are well-attested in quotations back to the 10th century; most explicitly, for instance, “The feastful day of the Epiphanye commonly called Twelf-day.”
, T. Becon Relikes of Rome , 1553.)
However (continuing with the OED), this doesn’t answer the question: twelfth-night did indeed properly refer to the eve of twelfth-day , i.e. to the night between the 5th and 6th of January.
(This is also attested back to the 10th century.)
So presumably at some point in comparatively modern times, the phrase twelfth-night lost its footing and slipped a day.
It’s easy to speculate on ways that could happen: maybe in eg the 19th century the actual twelfth-night festivities fell out of fashion and the bigger celebration was on twelfth-day, but for various reasons (eg the Shakespeare title) people were still familiar with the phrase twelfth-night , and so assumed that it referred to the festival they knew?
I suspect that some digging around in 18th/19th-century literature on Google Books might well be fruitful in narrowing down this sort of speculation, pinning down something of when and how the shift actually happened.
english.stackexchange.com/users/2587/PLL
No method.
The twelfth day is January 5th: 25-Dec 26-Dec 27-Dec
28-Dec 29-Dec 30-Dec
31-Dec 01-Jan
02-Jan
03-Jan
04-Jan 05-Jan January 6th is the day after the twelfth day of Christmas.
I am not an expert on this subject, but this is from the Wikipedia page on Epiphany : Christian feast celebrating the appearance of Jesus Christ to the Magi (representing the world.
, traditionally celebrated on January 6, the day after the twelfth day of Christmas .
english.stackexchange.com/users/212/b.roth
Linked on from the Wikipedia page , The Telegraph says,
But many people believe Twelfth Night falls on Jan 6, at the end of the 12th day after Christmas...
The difference in opinion is said to be down to the fact that in centuries past, Christmas was deemed to start at sunset on Dec 24 and
so the 12th night following it was Jan 5.
Nowadays, people count from Dec 25 and
so assume Twelfth Night falls on the 6th.
As it says - the method that counts the 12th night following Christmas i.e. which starts counting from Dec 26th, ends with the Twelfth night on Jan 6th
english.stackexchange.com/users/1798/JoseK
Is the 25th of December the 'first day' of Christmas, or the 'zeroth' day?
According to http://en.wikipedia.org/wiki/Twelve_days_of_Christmas Boxing Day, or the 26th, is considered the 'first full day' -
so that would place the twelfth night on the 5th of January, but the twelfth day would be the 6th.
I wonder if this is at all related to the idea of the new day starting at sundown (for example, the Jewish Shabbat runs from sundown on Friday to sundown on Saturday).
english.stackexchange.com/users/3718/HorusKol
I would imagine the confusion comes from having people who assume day precedes night tacking an extra, incorrect, night to the end.
The Twelfth Night is January 5th, the last day of the Christmas Season before Epiphany (January 6th).
In some church traditions, January 5th is considered the eleventh Day of Christmas, while the evening of January 5th is still counted as the Twelfth Night, the beginning of the Twelfth day of Christmas the following day.
http://www.crivoice.org/cy12days.html Case in point
the argument in this article only makes any sort of logical sense if you assume "The first day" is the evening of the twenty-fifth, which would make "the twelfth night" fall on the 6th.
(Night coming after day, even though they defined the day as something which started at night.)
english.stackexchange.com/users/2025/mootinator
I know English has (or at least had) some strange usages of  eve  and  night , but I still can’t figure out how December 25 th  and 12 can be combined to come up with January 6 th .  (This stems from my annual rant about the twelve days of Christmas, namely that despite what modern commerce would have you believe, they come  after  the 25 th , not before.)
What method of counting puts Twelfth Night on January 6th?
english.stackexchange.com
<numbers><time><dates><proverbs><christmas>
english.stackexchange.com/questions/7649
english.stackexchange.com/users/1547/Marthaª
Twelfth Night falls on January the 6th because the first day of Christmas actually falls on Boxing Day, the 26th December. However, "Twelfth Night" originally referred to the eve before the Twelfth Day, and it appears that at some point, Twelfth Night slipped to the following evening. The twelve days of Christmas referred to the twelve days  after  Christmas, i.e. starting from the first day  after  it; the somewhat misleading phrasing  the twelve days of Christmas  to refer to these days, historically, only comes later. Hence the twelve days have always been Dec 26–Jan 6, the first ‘day of Christmas’ is not the same as Christmas day, and  twelfth-day  is a synonym for Epiphany.Twelfth-night  did indeed properly refer to the eve of the  twelfth-day , i.e. to the night between the 5th and 6th of January.  This changed over time and research would be needed to find out why.
Twelfth Night falls on January the 6th because the first day of Christmas actually falls on Boxing Day, the 26th December. However, "Twelfth Night" originally referred to the eve before the Twelfth Day, and it appears that at some point, Twelfth Night slipped to the following evening. Boxing Day (26th December) is considered to be the first full day.
There is no such method. There is no method of counting which puts Twelfth Night on the 6th of January.
I would imagine the confusion comes from having people any people believe Twelfth Night falls on Jan 6, at the end of the 12th day after Christmas..
I would imagine the confusion comes from having people any people believe Twelfth Night falls on Jan 6, at the end of the 12th day after Christmas..
The twelve days of Christmas referred to the twelve days  after  Christmas, i.e. starting from the first day  after  it; the somewhat misleading phrasing  the twelve days of Christmas  to refer to these days only comes later. Hence the twelve days have always properly been Dec 26–Jan 6, the first ‘day of Christmas’ is not the same as Christmas day, and  twelfth-day  is a synonym for Epiphany. Twelfth-night  did indeed properly refer to the eve of  twelfth-day , i.e. to the night between the 5th and 6th of January.   This changed over time and research would be needed to find out why. Many people believe Twelfth Night  falls on Jan 6, at the end of the 12th  day after Christmas.. The first full day of Christmas is the 26th December. The confusion could ariserom having people any people believe Twelfth Night  falls on Jan 6, at the end of the 12th  day after Christmas. n some  church traditions, January 5th is  considered the eleventh Day of  Christmas, while the evening of  January 5th is still counted as the  Twelfth Night, the beginning of the  Twelfth day of Christmas the following  day.   
According to  http://en.wikipedia.org/wiki/Twelve_days_of_Christmas  Boxing Day, or the 26th, is considered the 'first  full  day' -
There is no method of counting which puts Twelfth Night on the 6th of January.