How is a chair defined? Surely something as simple and fundamental as a chair should have a reasonable definition.
In the dining room, the table is certainly not a chair. Because you don't eat from a chair. But in the university lobby, sometimes I would sit on the table when talking to someone who's standing up. And maybe I decided to sit on the floor and eat off the chair (with plates, mind you), because I needed to hide from someone.
So what is a chair? Does the number of legs define it? Does its location in the house define it? Does anything else?
Well. The honest truth is that a chair is an abstract idea, and it is not universally and globally well-defined. At some times a chair is something like this, and at other times it could be a completely different thing. The important thing is that within the context of the conversation, if I say "chair" it will be clear what I meant by that term.
And now we finally approach the point I am trying to make. $\Bbb N$ is an abstract object. When I talk to a fellow set theorist, it is clear that when I say $\Bbb N$, I mean the set of finite von Neumann ordinals (often called $\omega$). When I talk to someone who's doing a lot of category theory it is clear, at least to him I guess, that $\Bbb N$ is the initial object in the category of free monoids. When I talk to someone whose main interest is analysis, she might have a clear and intuitive as what are the real numbers, and from that she will derive the meaning of $\Bbb N$ as the smallest inductive set.
Sometimes it will be beneficial and useful to include $0$ as a natural number. For example if a set is finite if and only if its cardinality is a natural number, then surely $0=|\varnothing|$ is a natural number.
Sometimes it will be cumbersome to include $0$, when I was to talk often about sequences that look like $\frac1n$ having to add "for $n>0$" everywhere is a terrible crime against yourself, the reader and the listener. So just excluding $0$ works better.
Since you brought up the Peano arithmetic axioms, it is perfectly reasonable to develop those in a language which include $0$, and in a language which does not include $0$ (in which case $1$ is the minimal element, and you need to modify a few axioms accordingly).
So what is $\Bbb N$? It is an abstract idea, which you can implement, or realize in many various ways. Some are more concrete, some are less concrete. But it is an abstract object, that you know satisfy some properties, and perhaps the most important one is that it has a linear ordering that (1) has no maximal element; and (2) every proper initial segment is finite.
Why am I choosing the order here rather than the arithmetic structure? Because from this order we can define (in the broad sense of the word) the arithmetic structure, and everything else. With or without $0$. And this order gives us a uniqueness property. Every two linear orders with these two properties are isomorphic (and the isomorphism is unique, too).
So it really doesn't matter what is the set $\Bbb N$, because there are way too many options for interpreting $\Bbb N$ as a concrete set. It matters that we can use induction on that set, and that we know there is at least one set like that.