I was about to say that I could think of few reasons why anybody would need to know the time of the equinox to the nearest minute, before realising a rather entertaining set of circumstances took place last night…
* Wed 20/03/2019, 21:58 UTC, March equinox.
* Thurs 21/03/2019, 01:44 UTC, Full Moon.
The date of the equinox is used in the calculation of the date of Easter. To quote Saint Bede (The Reckoning of Time, 725), “The Sunday following the full Moon which falls on or after the equinox will give the lawful Easter.”
As I understand it, the church mucks up the calculation by defining the equinox to occur at midnight on 21st, and further defining full moon to occur 14 days after new moon.
Perhaps the BAA should start a campaign to “get back to Bede” and celebrate the astronomically-correct date for Easter this weekend.
To answer your actual question…. my impression is that these times are calculated by the US Naval Observatory and printed in the Astronomical Almanac, and that basically any and every source that quotes times for the equinoxes and solstices get them from the AA. So, I think if 21:58 is what the USNO says, basically that’s the time that everyone will quote.
I’m not sure what algorithm the USNO uses. When I have tried to do the calculation myself — which I tried to do in order to put equinox times on my website, In-The-Sky.org, I struggled to get exactly the same tiime as the USNO. For last night’s equinox, I get 21:44 UT. A difference of 14 minutes is larger than I would expect, since I used the NASA DE405 ephemeris to get the position of the Earth and Sun to high accuracy. I suspect I screwed something up — quite possibly there was an inaccuracy in my correction for the precession of the equinoxes (which is necessary to get the Sun’s RA and Dec for the epoch of the equinox, not J2000 coordinates).