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.