Is September 21 the first day of Fall?

Answer 1

By one definition, summer in the Northern Hemisphere begins at the summer solstice, when the sun is directly overhead at the northernmost point (the Tropic of Cancer). This is on or around June 21. In the Southern Hemisphere, the summer solstice occurs when the sun is directly overhead the Tropic of Capricorn, around December 21.

"Meteorological summer" in the Northern Hemisphere begins on June 1 and ends August 31.

Answer 2

No.
The Autumn equinox falls on the 23 of September. It is on this day that daytime and nighttime are of equal length. Ideally equinox's and solstice's are in the middle of their named seasons
Meteorologically the official start of Fall is the 1st of September (we do not feel the seasonal effects until some time after this date due to the thermal latency of the ground and sea) and continues until the 30th of November.
In the UK (where I live) the temperature has slowly fizzled down over the past few weeks, but there has been little shedding of leaves from our trees.
In conclusion, the official start of autumn is no real indicator of when it really starts. The signs of Fall arriving, move a couple of weeks this way or that each year.
So officially and anciently, we are already in the thick of Fall.
But so far as my senses are concerned, Autumn has only just begun...

Dr Gman