Teachers: Let's do it by the book and come up (somehow) with conflicting answers.
Mathematicians: Yeah, sorry guys. We made it all up.
Pretty much captures most mathematicians I know.
Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt.
It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set.
But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.
However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.
Now, mathematicians do throw the usual definitions away all the time. It's important to know when to reuse other people's coding, or roll your own.
In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were.
One enduring mystery in this vein was whether or not Euclid's fifth axiom was actually a logical result of the first four: https://en.wikipedia.org/wiki/Parallel_postulate#History
It's mysterious because it is only clear in retrospect. The fact that you can negate the parallel postulate and get a system that is still self-consistent is incredibly mysterious. This is what sets mathematics apart. We "make it up" like other human accomplishments, and yet we can't actually just make it up. If we arbitrarily defined 0^0 = π, we'd just be speaking nonsense.
It's kind of like this. A biologist independently discovers directed graphs, but refers to the nodes as "organisms" and edges as "parenthoods", and then spends five hundred years trying to prove that every organism has only finitely many parenthoods, thinking it must be true since it's obvious biologically! Without realizing that graphs are far more general and apply elsewhere.
Arbitrarily defining 0^0=π would not be speaking nonsense, it would just be speaking a little arbitrarily. Nonsense would be defining 0^0=rainbow
y = lim_[x->0] x^[a / log(x)]
We have log y = lim_[x->0] (a / log(x)) log(x) = a. So 0^0 equals any number at all! Of course, nobody in their right mind would define exponents this way, but the indeterminacy is inherent in the definition of the symbols.