Well, the difference you cite is the difference between Astronomical reckoning and Meteorological reckoning.
Astronomical reckoning is based on the tilt of the earth relative to the sun: Spring and Autumn start as the earth's tilt has the sun striking directly onto the Equator, Winter and Summer start as the earth's tilt has the sun striking directly onto the Tropic of Cancer and Tropic of Capricorn. (And, yes, all seasons are reversed in Southern Hemisphere vs. Northern Hemisphere.)
Meteorological reckoning is based on the (full) months when temperatures *normally* most reflect the season. Thus (in Northern Hemisphere) "Meteorological Summer" is June/July/August, "Meteorological Winter" is Dec/Jan/Feb, and spring and fall filling appropriately.
So, regionally, some cultures may be more inclined to refer/default to Astronomical reckoning and other regions/cultures refer/default to Meteorological reckoning.