As for the Gregorian calendar, the source is indeed (Ecclesiastical) Latin:
Deinde, ne in posterum a XII Kalendas Aprilis æquinoctium recedat, statuimus bissextum quarto quoque anno (uti mos est) continuari debere, præterquam in centesimis annis; qui, quamvis bissextiles antea semper fuerint, qualem etiam esse volumus annum MDC, post eum tamen qui deinceps consequentur centesimi non omnes bissextiles sint, sed in quadringentis quibusque annis primi quique tres centesimi sine bissexto transigantur, quartus vero quisque centesimus bissextilis sit, ita ut annus MDCC, MDCCC, MDCCCC bissextiles non sint. Anno vero MM, more consueto dies bissextus intercaletur, Februario dies XXIX continente, idemque ordo intermittendi intercalandique bissextum diem in quadringentis quibusque annis perpetuo conservetur. (Inter Gravissimas, 9).
Inter Gravissimas is the papal bull that established the current (i.e., Gregorian) calendar.
English translation by Spencer (1999):
Then, lest the equinox recede from XII calends April [March 21st] in the future, we establish every fourth year to be bissextile (as the custom is), except in centennial years ; which always were bissextile until now; we wish that year 1600 is still bissextile; after that, however, those centennial years that follow are not all bissextile, but in each four hundred years, the first three centennial years are not bissextile, and the fourth centennial year, however, is bissextile, so the years 1700, 1800 and 1900 will not be bissextile. Assuredly, the year 2000, as with our custom, will have a bissextile intercalation, February will contain 29 days, and the same rule of intermittent bissextile intercalations in each four hundred year period will be preserved in perpetuity.
Both the original Latin text and English translation (and also one to French) may be found here.
Here we see both bissextus and bissextilis. It seems to me that the former is preferred for the day and the latter for the year. (For this to be true, the translation of the relevant portion of the first sentence would be something like we establish that there must continue to be a leap day every four years.)
Addendum: As pointed by Vincenzo Oliva and tchrist in the comments, it is worth noting that bissextus literally means twice-sixth, as it was originally used to designate the intercalary day in the Julian calendar, that Julius decided to be the repetition (hence bis-) of the sixth day before the kalends of March (i.e., the 24th of February).