Simply put, our modern calendars and media have gotten it wrong. Somewhere along the line of history, perhaps as we became less rural and more urban as a society, the accurate meanings of Midsummer and Midwinter have been lost. One has only to consider the fact that the Solstices mark the MID POINT of the seasons via the course of the Sun. Basing calculations on four seasons, Summer would begin around the first of May, hence the old May Day celebrations, marking that very thing. This is the period when the planting is completed and the Earth is again filled with growing foliage. The Sun is moving toward its' annual apex or high point, which falls at Midsummer around the 20th of June, marking the longest day of the year. After that point, the Suns' energy begins to wane as it reverses its' course back toward Winter and the days grow shorter. (Hence, to say Midsummer is the beginning of Summer is contradictory.) The Autumnal Equinox marks the manifestation of Fall, when the hours of day and night are equal, though it could be said Fall begins around the first of August, when the nights start getting cooler and the foliage begins to dry up. The Harvest season also marks the beginning of the Autumn Tides. Winter as a season begins on the 1st of November, which is why 31 October was thought of as a time of death (the "dying" of the Earth until the next Spring), and came to be associated with celebrations such as Samhain, and later Hallowe'en. Around the 20th of December, we reach Midwinter, or the middle of the Winter season, at which point we experience the shortest day of the year and the Sun then begins to wax in energy as the hours of daylight again grow gradually longer. This explains the Midwinter celebrations (Yule, Saturnalia, etc.) which later became Christmas, as it marks the return of the powers of light in the midst of the deepest darkness. Winter begins to lose its' grip around Imbolc (now Groundhog Day in the USA), when the first stirrings of life begin to reappear, though there are technically six weeks of Winter remaining. The Spring Equinox ushers in the astronomical Spring season, as the hours of daylight and darkness are again in balance, as indicated by the old Spring celebrations which led to our modern Easter (a term based on the name for an old Celtic/Anglo Goddess of Spring). And we soon find ourselves back at May Day, when Summer actually begins.
Of course, in the Southern Hemisphere this is all reversed, as their Midsummer is in December and their Midwinter in June.
In our modern world, we tend to focus on the seasons as they apply to us or our particular region, and not on the astronomical reality which few pay attention to these days. Schools generally let out around the first of June, and many families take vacations during the three months we've come to think of as Summer, so it's understandable as to why we got so confused as to Midsummer and the beginning of Summer. Sadly, not many seem interested in correcting this error, as I have contacted several media sources (including the Old Farmers' Almanac), all to no avail.