From the article's example the simplicity of the binomial formula is extremely useful compared to a formula that would have to account for the case where k=0. Another commenter pointed out the useful elegance of 0log0=0 for physicists. These are the real world applications for mathematicians choosing definitions directly. Saying that an idea may be defined in many ways is correct, but choosing a working definition for the system helps to apply the definition to appropriate concepts. This is what mathematicians are doing when choosing a specific definition instead of saying that any definition will work.

One could argue the entire field of Real Analysis was formed because Calculus showed the world that we didn't really have those definitions, but they were needed. There are cases where the integral of the derivative does not equal the derivative of the integral (violation of the fundamental theorem of calculus) without having a specific epsilon-delta definition of limit.

Also, zero is not always the same as nothing or none. Zero is an abstract number that some have decided is useful to represent nothing or an empty set, but really comes from an abstract idea that you can count nothing and have a number. This goes back to the fact that numbers are pretty useful ideas regardless if you may consider one of them a function or not.