Its all arbitrary depends on how you define it.That would make sense just as long as there's three months between the end of the three hottest months and the start of the three coldest. Say June July and Aug are the three hottest, what happens if the three coldest are November, December, and January? That'd presumably give a meterological Spring of Feb March April and May and just an Autumn of October and November would it not?
And isn't it all kindof arbitrary anyway given that some months are 31 days long, some 30, and another 28/29.
Think it's easier to suggest that Feb marks the start of Spring whereas really the 91ish days between the darkest 91 days of the year and the brightest 91 days of the year probably starts around 6th of Feb-ish? So the start of Feb is probably the best rounding up approximate if you really do feel the need to tie it into the start of a particular arbitrarily sized month.
The animal kingdom doesn't bother with callendars and days of the week, but they, along with vegetation, seem to know when things are starting to get brighter and warmer, and it's stood them in good stead for millenia.
I presume it presupposes, 4 seasons of equal duration. Which is probably true when measures over a few thousand years. we all know about long winters and short summers though bit for expediancy 4 seasons of equal length. In Africa AFAIK they have a rainy season and a non rainey season. its not universal. in the southern hemisphere Summer is Nov, Dec and Jan. the point being its not universal. its all arbitrary and it depends on how you want to define it.
I presume at some point climate change may alter the definition somewhat but again it'll be arbitrary