It depends on where you are coming from. If your context are ordinal numbers then you are right and a typical definition of 0 is {} and of s(x) is x∪{x}. But if you working with finite fields for example then you only have an addition operation. "Successor" does not make much sense there, since 1+1+…+1=0 for the right amount of additions (you are calculating modulo a prime). Since 0+1:=1 is trivial you usually start with 1 and define 2 as 1+1, 3 as 1+1+1…

You don't usually define 2 at all when talking about abstract structures (it makes little sense to call the polynomial 2 as being "the 2" of the field of rational functions, for example). 2 is something that exists only in N, and talking about it in other structures makes sense only when you are referring to a ring homomorphism Z->F or something similar.

That definitely is a more clean way to look at the matter, yes. Nonetheless I've seen the definition I stated above a few times and the merit is that you do not have to take an implicit indirection every time you state something like "2≠0". And as a ring homomorphism ℤ->R for any ring R with identity element is already completely defined and in effect identical to the definition "2:=1+1…" for every positive whole number, it is really just a different way of formulating the same idea.

My point is that when talking about the definition of 2, it is enough to restrict ourselves to the natural numbers - since that is where 2 is coming from. If we then want to extend this symbol to other places - the meaning of such an extension will be given by a map, not by requiring a different definition. That is - there is only one 2, everything else is ψ(2) for the homomorphism ψ:Z->R.