What method of counting puts Twelfth Night on January 6th?

No method. The twelfth day is January 5th:

1. 25-Dec
2. 26-Dec
3. 27-Dec
4. 28-Dec
5. 29-Dec
6. 30-Dec
7. 31-Dec
8. 01-Jan
9. 02-Jan
10. 03-Jan
11. 04-Jan
12. 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.

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

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).

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.)