I find the "solstices/equinoxes mark starts of seasons" a bit foreign too, but… weather-wise, annual top and bottom temperatures are of course offset from the solstices due to thermal inertia.

In Finland the traditional division is that winter is Dec-Feb, spring is Mar-May, summer is Jun-Aug, and autumn is Sep-Nov. Historically it has made perfect sense, weather and climate wise – particularly from the point of view of agriculture, which is of course the reason people used to think about seasons in the first place!

February in particular is 100% winter in Finland with no signs of spring besides the days starting to get very noticeably longer by then. It's often the coldest month of the year and when schools usually have a week-long winter break. Similarly, August is very definitely a summer month except in the far north where spring comes late and autumn early. The academic year in schools and universities typically starts at the end of August, so that's a clear and important dividing line in many people's lifes. In Southern Finland, December is these days rather autumny more often than not, and there's often no lasting snow until January (if even then). June is a crapshoot, it can be nice and warm or surprisingly cold.

I guess Jan-Feb are definitely winter, Apr-May definitely spring, Jul-Aug definitely summer, and Oct-Nov definitely autumn. The rest are kind of transitional and their weather unpredictable. Of course, the climate change isn't helping things, either.