Boring weather fact. We consider june-aug as summer because having seasons designated by 3 full calendar months makes it possible to compare with past records which were not as detailed as they are now (eg wettest spring since 1853 type comparisons). Hence spring is mar-may, summer jun-aug, etc.
If we were to use more detailed stats, to ignore the need to consider seasons as being whole months, and indeed why do they have to be equal lengths…. we’d probably designate summer as being late June until maybe mid-late September. June is typically warmer but wetter, but in terms of the weather being generally drier and sunnier, Sept wins.
YMMV, depending also where you are, but this bears out my observations in the SE of England. I’d be far more likely to plan a camping trip in Sept compared to June.