Although all the above comments are correct, given the time-frame of the traditional songs, nevertheless, the 8th of January date is technically correct, too. Keeping in mind the dating corrections, it goes like this: (Source Duncan Emrich, L of C Anglo-American Songs & Ballads [AFS L14])

Per "old style" reckoning, old Christmas became Jan 5 from 1752 to 1799. In 1800 another day dropped and it became Jan 6; in 1900 the same, to Jan 7. I haven't confirmed this but the convention should be that old Christmas is now Jan 8 and the song suddenly becomes correct.
(BTW, for UK & colonies, last day Julian (OS) was Sept 2, 1752; The following day was Sept 14, 1752 Gregorian Calendar (NS)