This is foosball.
The sport with the egg is "gridiron football", we call it "football" for short for the same reason other countries call association football "football" for short, it's the most popular variant here.
We don't call it "American football" because that's not what it's called, you lot say that because of point 2. Nobody is going to say "European football" because that's not what it's called either.
The word "soccer" was imported from the UK (seriously that word is british as hell, you really think we came up with that shit?), and we use it for reasons outlined in points 2 and 3, and also because "european football" is inaccurate, tiresome, and stupid sounding.

~~5. This is the first and only time I've ever heard of rugby union football being referred to as "ruggers", I never thought I'd read a dumber-sounding word than soccer but you're just full of surprises today.~~ For some reason i was imagining one of the replies was the same person, this was in error.