After the initial comments, I started doing my own research to try to find better evidence of this use. I considered adding data to my question, but it seems this may be more appropriate in an answer.
First, I'll say that I skimmed roughly 100 hits in Google Books for the phrase "bissextile day" in the 20th and 21st centuries, and I only found two examples of this phrase referring clearly to February 29th, a novel from 2008 and a pop history book from 2015. Many other uses clearly make reference to the traditional February 24th definition. The remaining minority are ambiguous, using the word without formal definition, as in "the year added to leap year" without specifying a date. I can't find any clear reference to February 29th as "bissextile day" in the 20th century, nor in the roughly 40 hits in Google books from the 19th century. (I was going quickly, so I may have missed something. But if this usage exists, it must be in a small minority of cases.)
Therefore, at least in written works, the use of "bissextile day" to refer to February 29th is much less common than the traditional use. (I would note, however, that as of the time of this answer on February 29th, there are over 40 uses of the phrase on Twitter implying a reference to today, and many more uses of the word "bissextile" also in reference to the day rather than the year.)
Notably, many of these tweets make explicit reference to online references, which for some reason seem to privilege the February 29th interpretation (despite the fact that I can't seem to find any significant evidence of it in written works).
A few citations in reference sources that seem to indicate this (new?) sense:
Merriam-Webster is highlighting this entry today:
A leap year is a year that has an extra day—366 days, with February 29
as the extra day. It has another name in English: bissextile year
(and leap day is also known as bissextile day).
(The link also explains the etymology relating to February 24th, but this quotation implies that the "extra day" February 29th is "leap day" and can also be called "bissextile day".)
The current Word of the Day at Dictionary.com is bissextus, whose definition is given as:
February 29th: the extra day added to the Julian calendar every fourth
year (except those evenly divisible by 400) to compensate for the
approximately six hours a year by which the common year of 365 days
falls short of the solar year.
Webster's New World College Dictionary with etymology about February 24th, but first definition:
- denoting the extra day (February 29) of a leap year
Wordsmith.org includes a usage from 2004 and the following etymology implying bissextus = February 29:
[From Latin bisextilis annus (leap year), from bissextus (February 29:
leap day), from bi- (two) + sextus (sixth), from the fact that the
sixth day before the Calends of March (February 24) appeared twice
every leap year.]
Macmillan Dictionary with a 2008 citation of usage from the Wall Street Journal and this explanation:
Contrary to what you might think, bissextile has nothing to do with
issues of sexual orientation or gender! It simply refers to a leap
year – a year containing February 29th and therefore a total of 366
days. A noun derivative bissext (with a variant bissextus) refers to
the extra day itself, though leap day is a common alternative in much
I don't know how to interpret this. It's unclear what the basis is for all of these online reference works implying a meaning that doesn't seem standard.