No it doesn't. The summer solstice is called the summer solstice because it occurs during the summer, not because it signifies the start of the summer. What it signifies is the day with the longest period of daylight. As far as I know it has never been explicitly connected to the start of summer. Despite the astronomical summer often starting at the same time. It has been connected to midsummer though in many places, especially in the days of old. That is the middle of summer. And if anything it makes a lot more sense to be connected to that. That by the way is the solar summer take. But either way the meteorological summer makes the most sense to me.
Either way, today it is summer in all 3 takes. At least in the northern hemisphere.