In India, by summer solstice, large parts of the country are already under the sway of Monsoon rains. And so technically, the peak of summer is past.

https://en.wikipedia.org/wiki/Monsoon

It's not about culture alone, rather also about the weather patterns. In India, monsoon is the major climactic event around then. So it's not spring-summer-autumn-winter in India (in most part of the country), rather, it is summer-monsoon-winter

Not even Western in general. In New Zealand summer starts three weeks before the solstice, on December 1. Each season is three full calendar months.