The Ancient Egyptians had three seasons, and they based their calendar around the Nile's flood cycle. These were Akhet in September-December (when the Nile flooded), Peret in January-April (when they'd plant their crops), and Shemu in May-August (when they'd harvest the crops). Since the Egyptian economy relied heavily on agriculture, and the Nile floodplain was the only arable land in the country, they based their year around the flood cycle. A happy Shemu to you, by the way!

I don't know about common, but in Vedic writings they speak of six ritus, which roughly correlate to seasons: spring, summer, rain, autumn, fall, winter.

I imagine in many places with a very dry part of the summer followed by heavy monsoon rain you would find a seasonal cycle described as having at least five seasons.

FYI, some consider the solstices and equinoxes to be the midpoint of their respective seasons, not the divider. Hence, midsummer is not halfway between the summer solstice and the autumnal equinox; it's around the solstice.

Astonomically, this makes sense, but some places, e.g. the US, experience significant lag in seasonal temperature change, so the seasons are considered to start a half-season later than they do astronomically.