If you want to go down the prescriptive path, a more correct word relative to "autumn" is "hærfest". Just like "fall" replaced "autumn", "autumn" replaced "hærfest".
Meanwhile, "autumn" is a loan word from French. "Autumn" and "fall" were both used in Britain, but "autumn" eventually won out (after the colonization of North America).
"Fall" (along with its counterpart "spring") was commonly used in Britain when America was colonized. The British later stopped using "fall" (although they did keep "spring") and switched completely over to "autumn". But Americans never made that switch and continue to use both terms just as their British ancestors did.
"Fall" never replaced "autumn". It replaced "harvest" (just as "spring" replaced "lent"). "Autumn" was borrowed from French, and then "fall" and "autumn" existed side by side for hundreds of years in Britain, up until very recently when "autumn" replaced "fall".
not sure how is that relevant?