And I don't think it was after the fact, or a generalisation of anything, to say that the natural numbers are the classes of sets with their own cardinality. It was the definition given by Frege, who I believe was the first person with the audacity to define something so primitive. And defining natural numbers in terms of their cardinality strikes me as entirely natural, once you have it that natural numbers are just the finite numbers you use to count stuff (i.e. cardinal numbers --- counting stuff by forming one-one correspondences).
Frege's definition is not the one given by mathgrad, because it doesn't work in ZFC. We can't use equivalence classes, so instead, we pick out canonical representatives of the classes, which is a good enough compromise.
That said, I don't think there is one definition of natural numbers to rule them all. I am personally quite fond of the Church numerals, where a natural number n is defined as the higher-order function which composes its argument with itself n times. I prefer the thought that what counting is about is the repetition of a single operation.
As for the rest, I am the spineless type of formalist who is happy to throw his hands up in the final analysis and say that all of mathematics is arbitrary. And so unlike the OP, I won't draw a distinction between 0^0=1, 1+1=2 or 0.999...=1. You can mangle all of these as much as you like, but so long as you keep the expressive power of arithmetic, you'll find an analogue of Turing equivalency to translate to the original.
To run with your example: why must a definition of natural numbers have to involve sets of certain cardinality? You're telling me that because some people came up with a more general definition for natural numbers after the fact, that makes one definition of natural numbers more 'natural' than another.
I claim you're just reinforcing my point: the definitions were chosen to make natural numbers a specific instance. Everything here is chosen in some way to further some goal (usually mathematical aesthetics), and eventually you get down to the bottom and what do you have? A bunch of definitions that one may choose to use or not.
So whether I want to impose 0^0 = 1 by fiat is equivalent to whether I want to assume enough foundations of category theory to prove 0^0 = 1 in that system. You (and mathgrad) are just under the spell that because there is a whole lot more mathematics floating around (and big words and important people working in those fields) it somehow makes the latter less arbitrarily motivated by a desire to make theorems work out nicely. It's certainly pleasing that it does, but that doesn't make it somehow deeper or more natural than a different formalization in which 0^0 = 0. It just serves a different purpose.
This is the entire point of the OP and the original comment: you can generalize the meaning of 0 and 1 and a^b to make it suit your need to express certain theorems and patterns, and that is a key part of the power of mathematics. Whether it's a "truth" or an "axiom" just depends on how far down the rabbit hole you're willing to go, and the distinction is irrelevant because as far as deciding what 0^0 should be they're equivalent.