There is no term “American Soccer” so by default it is best to use football to mean the oblong pigskin, and soccer to mean that round black and white ball kids use around the world.
The fact that there is no term ‘American Soccer’ is completely irrelevant. The fact that you (plural) chose to misname your sport after ours, despite the fact Gridiron is much closer to Rugby than football, is none of our concern, and does not mean we must stop calling our sport what we’ve always called it to make way for your mistakes.
It’s a bit confusing attempting to conduct the same argument over two threads, so I’ll focus here on the English language, and regarding the Football/American Football debate, in the Soccer thread (the changing of the title made me lose any confidence I might have had in the moderators).