Probably depends on where you are, etc., but as an European, I was taught in school two ways of splitting the year up into seasons: calendar/astrological and meteorological. Calendar split is based on solstices and equinoxes (21st March, 21st June, ...), whereas meteorological is based on month start (1st March, 1st June, ...). They use this also in weather reports, for example, where on 1st March they would add "Today starts meteorological spring" and on 21st March "Today starts calendar spring".