Autumn starts on March 1 etc.
March 1? Now that is weird. Shouldn't it be on an equinox?
Not necessarily. It would make more sense for "midwinter" to actually be in the middle of winter.
A couple things about this. First, you are talking solstice, whilst I was talking equinox. Not super relevant. Second, 'midwinter' is a day which has colloquially been applied to the day on which the solstice occurred, but I certainly wasn't advocating such a term, nor do solstices need to be tied to them by any means. Third, it's a semantic argument - I mean, what defines winter? One could argue that winter is defined such that the astronomical event stands in the middle of the season. Or you could talk about it being cold. So, the thing you have to remember is that whilst the days get shorter or longer in relation to the movement of the sun, the shortest day of the year is not the coldest (well, rarely it is by fluke, but...). It's the same reason that the hottest part of a day is generally in the afternoon, and not at noon. It may be maximally heating at noon, but it continues to heat up for a bit afterwards.
Also remember that the Celtic culture, from whom we originally took the terms midsummer and midwinter, reckoned the seasons differently, with the feast days that they (and neopagans) called Samhain (the origin of Halloween), Imbolc (the origin of, of all things, Groundhog Day), Beltane ( related to May Day) and Lughnasa (near 01 August) being the start of their seasons. In that context, Midsummer (summer solstice, ~20 June) is very close to the actual middle of summer (15 June by this reckoning). Many other European cultures of the middle ages reckoned the season in the same way. Some in China, Japan, and Korea reckon the seasons similarly, though with the season changes falling somewhere between the 5th and the 10th of each month, making the summer solstice even closer to midsummer.
Meanwhile, meteorologists use yet a different definition, the one Jimmmmm was talking about: spring starts on 01 March, summer on 01 June, and so on.
Meanwhile meanwhile, temperatures in ANZ and southern S. America, due to their climates, have hot seasons that start earlier--hence, they usually use 01 June for the start of summer.
So, to recap:
~01 May: start of summer in the Middle Ages and neopaganism
~05 May: start of summer in parts of SE Asia
01 June: start of summer for meteorologists and ANZ residents
~20 June: start of summer for astronomers, North Americans, and most Europeans; traditionally, midsummer.