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The Artscroll in Taanis 10a note 42 states that
...according to halachah September 21 is not reckoned as the day of the equinox. Rather, the equinox occurs (in this century) on October 7th or 8th.
What is the reason/source for this, considering that Chazal were aware of the 365-day solar calendar? And what does it mean "in this century," seeing as the equinox should ostensibly be the same from century to century (based on my limited knowledge of even basic astronomy)?
The answer comes from a complicated discussion of the different calendar systems. Put simply, due to shifts in the calendar viz a viz the seasons (because the earth doesn't orbit the sun in exactly 365.25 days) Pope Gregory XIII in 1582 decided to advance that year's calendar by 10 days, so October 4 was followed by October 15 in the newly created Gregorian calendar. Although the Gregorian calendar essentially has become the universal standard, Judaism doesn't follow Pope Gregory or his calendar, even for solar-related matters (since regardless of this innovation, Judaism has a lunar calendar). This means our reckoning of when the equinox is follows the old calendar set-up (for arguments sake, the Julian Calendar).
Regarding "this century", this refers to how the Gregorian calendar is set up differently than the Julian. Normally every four years there's a 29th day of February. The Gregorian calendar added a caveat that every 100 years doesn't have an extra day (1700, 1800, 1900 didn't have), but every 400 years does (2000 had). Again, we don't follow this convention, so in 2100 the two calendars will drift apart by another day.
In summary, the Jewish reckoning of the "solar calendar" is currently 13 days out of sync with the Gregorian calendar, and every 400 years will drift apart another 3 days. Although, according to this, the fall equinox should be October 4th, not 7th like Artscroll says. Not sure...
Also see here.
Managability.
There is an irrational number of days in a year. It's one of those numbers like the square root of 2, or pi, that cannot be written down. So, every calendar is based on an estimate.
The Jewish calendar uses a pretty precise one, which we call Tequfas Rav Ada (the seasons of Rav Ada) -- 365 days, 5 hours, 997 chalaqim, and 48 rega'im. (A cheileq is 1/18 of a minute, i.e. 1080 per hour, and a rega is 1/64th of a cheileq.)
Which allowed Sanhedrin to make a calendar, but for something like davening, it has to be within the reach of everyone. And really, do we need to-the-day precision for when the rainy season starts? It's not like seasons actually change on the moment of equinox?
So for the purposes of the siddur, they decided to go with Shemu'el's simpler, if less accurate, estimate -- a solar year is around 365-1/4 days. In Israel, it was particularly easy -- the Romans were using the Julian calendar, based the same estimate! So there would be a simple rule for when to start, based on the secular date. But even in Babylonia, in the Sassanid Empire, the calculation was more manageable.
Now that most of the world uses the Gregorian calendar, that simple rule got a little more complicated. But since the change is only once a century, three centuries out of four, life is still pretty easy.
Every year, the calendar is a quarter of a day off, so the Julian calendar adds a leap day when that error comes to getting the date wrong. But, they start days at midnight. We start days around six hours earlier (a quarter of a 24-hour day), at sunset. So, the way we count days, the error adds up to the next day a quarter of a cycle earlier. So, the rule for "Vesein Tal" is that it's a day later the fall before a leap day.
But again, we aren't really basing ourselves on the Julian calendar. It's just a way to make the rule easy. We are basing ourselves on Tequfas Shemu'el, which just happens to be the same 366-1/4 day estimate for the year as the Julian calendar is based upon. Balancing ease of use and the kind of accuracy the tefillah requires to be useful.
Taken from https://torahclarity.blogspot.com/2021/12/fixing-jewish-calendar.html
The Lunisolar Calendar
It is well known that a year consisting of twelve lunar months is roughly eleven days shorter than the solar year, and that the Jewish year sometimes contains a thirteenth month in order to address this imbalance. The Torah says that Pesach is in 'the month of the spring,' Shavuos is the 'harvest festival' and Sukkos is the 'festival of ingathering (of crops)', and this could not be maintained using a uniform year of twelve lunar months.
The extra month of Adar Sheni is therefore added in seven out of every nineteen years. The resulting 235 lunar months in nineteen years produce an average year length of 365.2468 days; the actual solar year is 365.2422 days long.[9] The average discrepancy is about seven minutes per year, or a day every 216 years. Cumulatively, this has caused a seasonal drift of around seven days from the time that the calendar was originally fixed.[10] The spring equinox is astronomically on March 20-21, but by halachic calculations is currently on March 27-28.[11]
[9] The average year length produced by the Jewish calendar is thus more accurate than that produced by the Julian calendar (365.25 days), used by the secular world until 1582, but less accurate than the average year length produced by the Gregorian calendar (365.2425 days). It is interesting to note that were we to update the mean molad interval to a more accurate measure, correcting the inaccuracy in measuring the lunar months, the Jewish calendar would produce a mean year length of 365.2426 days and the inaccuracy of the current lunisolar calendar would also be virtually eliminated.
[10] The length of the solar year, measured in solar days (the rotation time of the Earth), has not changed significantly over this time (the length measured in fixed length seconds has changed by a similar degree to the change in the length of time taken for the Earth to rotate).
[11] These halachic calculations are codified by the Rambam in Hilchos Kidush HaChodesh, chapter 9.
