If you define 2 differently what's the definition of 2?

One popular definition (at least, among that small class of people who need to define 2) is { { }, { { } } }.

Another, less used nowadays, is { z : ∃x,y. x∈z ∧ y∈z ∧ x ≠ y ∧ ∀w∈z.(w=x ∨ w=y) }.

In surreal numbers, 2 is { { { | } | } | }.

When you write "1+1" you may mean two things: "the result of doing the adding operation to 1 and 1", and "the successor of 1". It just happens that we use "+1" to denote both of those. The fact that successor(1) = add(1,1) isn't completely trivial.

Principia Mathematica, though, takes a different line. IIRC, in PM "2" means something like "the property a set has when it has exactly two elements" (i.e., when it has an element a and an element b, and a=b is false, and for any element x we have either x=a or x=b) and similarly for "1" (with all sorts of complications because of the hierarchy of kinda-sorta-types PM uses to try to avoid Russell-style paradoxes). And "m+n" means something like "the property a set has when it it is the union of two disjoint subsets, one of which has m and the other of which has n". Proving 1+1=2 is more cumbersome then. And PM begins from a very early point, devoting quite a lot of space to introducing propositional calculus and predicate calculus (in an early, somewhat clunky form).

In type theory and some fields of logic, 2 is usually defined as (λf.λx.f (f x)); essentially, the concept of doing something twice.