"The season of the year between summer and winter, during which the weather becomes cooler and many plants become dormant, extending in the Northern Hemisphere from the autumnal equinox to the winter solstice and popularly considered to include the months of September, October, and November..."
From the Wikipedia entry for Autumn:
"Some cultures regard the autumnal equinox as "mid-autumn", while others with a longer temperature lag treat it as the start of autumn. Meteorologists (and most of the temperate countries in the southern hemisphere) use a definition based on months, with autumn being September, October and November in the northern hemisphere, and March, April and May in the southern hemisphere."
From Live Science:
"The autumnal equinox is the beginning of "astronomical autumn," which ends with the start of winter. In the Northern Hemisphere, that is from around September 22 to about December 22, though it varies from year to year, according to the NOAA. In the Southern Hemisphere, the season runs from about March 20 to June 21.
Weather forecasters and some countries define the season as "meteorological autumn" by the three months in which the weather changes. In the Northern Hemisphere, this is September, October, and November. In the Southern Hemisphere, meteorological autumn occurs in March, April, and May."
In other words, the astronomical definition, while cute and clear, is not necessarily the relevant definition...
The reason many Americans (and other people in temperate climates in the Northern Hemisphere) think fall begins at September is because that is what many people mean when they say fall -- that is how language works...it's important to recognize that the scientific meaning of many words typically is NOT the same as the general usage of the word....that doesn't make the non-scientific usage wrong, it simply underlines the importance of context, and in this context it is perfectly reasonable to think that by "fall" the OP (who is American) was meaning by the end of August or so...certainly the OP could choose to clarify their intended meaning of a contextual term that possesses multiple meanings, but it is by no error of the reader to think the intended meaning of "Summer" was by September....
The reason most people think it's the beginning of September is that is when school use to start
The very concept of fall as a season is grounded in local changes in temperature, and those changes occur at different times in different places...a rigid date of September 21st, quite frankly, is at odds with the very purpose of an inherently contextualized term...in other words, the September 21st definition is technically correct but hardly relevant in most situations....