terminology - Equinoxes and solstices: start of the season or mid-season?

This all depends on the context in which the word is used.

From an astronomical perspective, the solstices are midsummer and midwinter, and the equinoxes are the middle of autumn and the middle of spring.

However, the seasons, as perceived in terms of weather, lag somewhat behind the astronomical seasons (presumably this is due to the thermal mass of the planet?). For example, most people of my acquaintance think of northern-hemisphere summer as occupying approx. June/July/August, putting the June solstice near the start of it. Therefore, colloquially, the solstices and equinoxes seem to roughly match up with the starts of the seasons, and hence are often treated that way.

Why does this apply to the equinoxes more than the solstices? (citation needed - but purely from personal observation, many people think of the summer solstice as "midsummer" but the spring equinox as "the start of spring"). I'm speculating, but my guess is that midsummer and midwinter are really obvious to anybody keeping an eye on the sky (when the sun reaches its greatest or least north/south extent, or "when the days start getting shorter", or whatever). The equinoxes are slightly harder to judge, and so aren't noted in quite the same way.