Australia

Oh very well done, you found the one dictionary that limits the definition of ball to spherical objects. That, unfortunately, makes that dictionary wrong, because a dictionary's job is to describe language as it is used, and you'd be hard-pressed to find anyone in good faith who does not call the ball used in Australian football, American football, or the two rugby codes, a ball. Oxford does a much better job:

a solid or hollow spherical or egg-shaped object that is kicked, thrown, or hit in a game.

And so does (unsurprisingly, since it has the tendency to be the most complete source for a lot of words) Wiktionary:

An object that is the focus of many sports and games, in which it may be thrown, caught, kicked, bounced, rolled, chased, retrieved, hit with an instrument, spun, etc., usually roughly spherical or ovoid but whose size, weight, bounciness, colour, etc. differ according to the game

The history is actually interesting. The story you told is one I've heard before and at one point believed myself (though I've never heard someone take the inflammatory tone of calling it "cheating", so much as it usually being described as him being so wrapped up in the heat of the moment). But it's not quite right.

The truth is that prior to the mid 19th century many different forms of "football" were played across England, and whenever teams from two different areas wanted to play each other they would have to agree on a set of rules. It may have been sort of like how International Rules Football today is a compromise ruleset between Australian and Gaelic football. Then in the early to mid 19th century specific codes started to coalesce and become more standardised. Rugby has its first written standard ruleset in 1845, and what we know today as soccer followed shortly after in 1863 with the formation of the Football Association (from which soccer takes its name).

For a time between the formation of the FA and its first finalised Laws of the Game, rugby clubs remained members, but following a decision to remove the draft rules that would allow carrying the ball after "he makes a fair catch, or catches the ball on the first bound", rugby and soccer went their separate ways and eventually evolved into the sports we know today. (Incidentally, while I knew the information from the previous paragraph already, apart from specific dates, this whole paragraph was entirely new to me in looking up those dates just now.)

The use of the term "football" for all these sports, incidentally, comes from the fact that they are propelled forward on foot, rather than on horseback as in polo, or with a racket as in tennis. The origins of football sports are so intermixed it is impossible to say that one inherently has a better claim than any other. I would certainly not claim an Englishman is wrong for calling it football. But in this country, it has always been soccer, because we have our own local football codes.