Historically, the third season was referred to as 'harvest' to reflect the activity of the season when the farmers gathered their crops for winter storage. This period is traditionally between August and November. The astronomical season begins on the autumnal equinox and ends on the winter solstice in the Northern Hemisphere.
The word 'harvest' comes from the old Norse word 'haust' meaning to gather or pluck. But as society progressed and more people began moving into cities the term harvest fell out of common use and was replaced by the phrase 'fall of the leaf' to refer to the third season of the year when trees lose their leaves.
Over time the phrase 'fall of the leaf' was shortened to 'fall' because the full phrase became a bit clunky to use in daily conversation.
Surprisingly, the word 'autumn' has unknown roots but was used by writers as early as the 1300's. As such we can find it in the works of Chaucer and Shakespeare. Shakespeare refers to the seasons as "The spring, the summer, the chiding autumn, angry autumn". However, etymologists have not determined the precise origin of 'autumn'.
As English spread to the New World the common season names split as well. The use of the word 'fall' fell out of favor in England. So today American English uses the word 'fall' while British English uses 'autumn' almost exclusively. So that explains why in referring to the daylight savings time change we use the phrase 'Spring ahead and Fall back'. I want to wish you all a safe and prosperous 'third season' be it fall or autumn.