The mechanics of astronomical time has always made me MEGLO, but yet I still want to know stuff. We have a local solar day defintion of new year, simply 12 midnight in our time zone at the Jan 1 transition.
However, the year can be precisely defined in terms of the earth's *orbit*. I once thought that precession of the spin axis would skew the year, and in 12,000 years or so, it would be cold in July , but then I learned that no, the defintion of the year takes precession into account -- it's got to do with where the earth's axis is pointing relative to the orbit. The US Naval Observatory has this down pat I'm sure.
So, the year has to do with completing one orbital cycle where the earth's axis "comes back to the starting point". Now, however that works out, the center of mass of the earth will be at some point at that time, and our rotational position on the surface is irrelevant.
Thus, I would declare the "real New Year's" would be this exact time, adjusted to our local time zone.
Now, when is that? I suspect that leap year considerations (the nasty fact that the orbital cycle is not an integral number of solar day cycles) can can skew that back and forth over the 4 year cycle, probably up to 3/4 of a day. And then when you add local time zone, it might well occur on Dec 31st, all the way to Jan 2 or something.