Here in the UK, the Met Office use straightforward whole months: Autumn is September, October and November.
But many people would give September 21st as the start date, believing that that is the equinox, although in many years, including this one, it actually falls as late as the 23rd.
And then, of course, you have the phenological perspective, which looks at the natural world and such things as animal migration and changes in plants, leaf colour etc, and so is liable to far greater variation depending on region and actual weather conditions.
If summertime is over, surely now there can be no doubt that autumn has actually started?