In the Peano axioms, the concept of addition is described:
where $S(n)$ is the successor function. As far as I can tell $a+0=a$ is (even in books such as Tao's Analysis vol 1) not an axiom but rather an identity or a definition.
Why is this so? To my mind an axiom is something we accept as true without further proof, which seems to be exactly what $a+0=a$ is doing. I accept that it's an identity, but if we are to call it a definition as well, to me a definition is a sort of "relabeling" or "rewording" of something.
For example we might "define" $1 := S(0)$ or $2 := S(S(0))$ or even $2 := S(1)$ because all those nested functions become unwieldy. But my point here is that the "definition" is simply placing some new notation or syntax or label on something that we can already describe in other terms.
So if $a+0=a$ isn't an axiom we simply accept as true, and it is a definition instead, how do we describe the underlying operations in terms of the stuff we already have? What is $a+0=a$ formally "shorthand for"? Is it a matter of first-order logic? Functions?
For example I might think that $a+0=a$ is a way to say $a$ is shorthand for some arbitrarily-deep nested successor function stack and then $+0$ is equivalent to "not recursing any further" but I don't know if this is right or if this is even how it's formally described.
How do we break down $a+0=a$ even further? What is it a definition of? Why do we not call this an axiom as well?
Edit: To be clear, I completely understand that $a+0=a$ is the "base case" of an inductive concept $a+S(b)=S(a+b)$ -- but this is not what my question is about.