For "those of us looking in" (see parent^5, which this thread is about) it's not just pedantry but a necessary distinction. Although your game's True Name might very well be "football", to all of the rest of us "football" means Association Football (soccer). If you say "football", we'll instinctively think "soccer".
Every. Single. Time.
Sometime's we'll double-take, realise it's an American talking, and they probably actually mean "American Football". But it's not what we think of first, and if we're not sure if it's an American speaking (like you can't on the internet) then it may well not occur to us.