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According to the formal definition, solstice occurs when the Sun's
apparent ecliptic longitude is 90° or 270°.

http://asa.usno.navy.mil/SecM/Glossary.html#_S

Longitude increases at a nearly constant rate of 1° per day, or 1
arcsecond per 24 seconds of time. Thus it yields a more definite time
than you'd get from declination stand-still.

Similarly, an equinox is when the Sun's longitude is exactly 0° or 180°.
Its declination would be zero at these times if the Sun were exactly on
the ecliptic. But in general it's not, so normally the Sun is slightly
displaced from the equator at an equinox.

For example, I calculate the winter solstice at December 21 23:04:09
Terrestrial Time, or 23:03:02 UTC. JPL HORIZONS agrees within about 36
milliseconds of arc. Possibly the discrepancy is due to different
ephemerides or precession / nutation models. Or I blundered a little
with the JPL DE422 ephemeris and IAU 2006/00A model.

At that time, longitude and R.A. are equal. I believe this is always the
case at a solstice, because the applicable longitude and R.A. meridians
coincide.

It's not the case at the equinoxes, where the meridians are skewed by
the obliquity of the ecliptic (ca. 23.5°). This would make no difference
if the Sun were exactly on the ecliptic, but as I said above, generally
it's not. For instance, I put the autumnal equinox at Sep 23 02:30:12
TT. At that time, Sun geocentric apparent R.A. was a couple tenths of an
arcsecond short of 180°, and declination about .4 arcsecond south of the
equator.

The reason the Sun doesn't exactly follow the ecliptic is suggested by
definition #1 of "ecliptic" in the glossary above: "1. The mean plane of
the orbit of the Earth-Moon barycenter [center of gravity] around the
solar system barycenter." Since Earth and Moon orbit once per month
about their barycenter, in a plane not parallel to the ecliptic, an
imaginary observer at the geocenter travels north and south of the
ecliptic on a monthly cycle. Due to parallax he sees a corresponding
wobble in the Sun's declination. It's a few tenths of an arcsecond.

   

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