Very true. Soccer was an original term because Rugby was also known as football. The term 'Soccer' only fully fell out of favour after the 70s by memory.
It's the same in Australia. The term 'football', I think, applies to Aussie Rules.
Partly true - soccer comes from posh nobs liking to shortening things - rugby football becomes ‘rugger’ and association football becoming ‘socca’ which evolved into soccer, so term is English not American