It would require more research than I'm willing to do, but the only part of that article that set off my sports-history-nerd Spidey Sense was this:
In full, it was known as gridiron football, but most people never bothered with the first word.
I don't know that anyone actually involved in playing or codifying the game ever used "gridiron football" in anything like the same official way that Association football or Rugby football were used. It feels much more like outside observers trying to impose logical categories from afar, British exceptionalism at its finest. AFAIK, gridiron was always used as a nickname for the field, and the sport itself was only ever widely referred to as "football," American exceptionalism at its finest.