Set theory does not answer the question, "what is two itself?" This is a common misconception. It just provides a way to translate statements about the natural numbers into statements about sets, which then you can use for proofs and formal constructions. It's an encoding, not a definition. In particular, the definition you give of two is not correct and I'd be interested to learn who told you it.
Further, the "modern" way to discuss this is not categorification, despite what Baez says; people who work and publish on the foundations of mathematics almost universally do so in the language of set theory.
Since sets are 0-categories, we can't escape set theory when talking about structures like the natural numbers. A natural numbers object is a feature of a topos, preserved by topos functors (geometric morphisms). Nobody told me this definition; it's something I had to absorb for myself when learning topos theory.
Formally, let N be the natural numbers object (in some topos), let z : 1 -> N be the zero arrow, and s : N -> N be the successor arrow. Then z;s;s : 1 -> N is the arrow which chooses 2 as an element of the NNO. Since z and s are unique up to unique isomorphism, so is 2. Moreover, since geometric morphisms between topoi preserve finite limits, the NNO should also be preserved, and that includes 2.
When the topos we choose/define is (equivalent to) Set, then we get the standard ordinal-number definition of 2.
To use a pun, this lets us upgrade from Dedekind-categoricity to a more modern and natural sort of categorical categoricity.
Of course we can escape set theory (and categories) when talking about the natural numbers. The concept of "two" predates the concept of a set by at least a thousand years. People were happily manipulating and investigating the natural numbers before set theory ever came along.
As I said previously, it is true that set theory provides a way to encode the natural numbers as sets (or features of a topos, etc.), so that questions about natural numbers can be stated as questions about sets. It is further true that this endeavor can be incredibly fruitful, for instance for studying the foundations of mathematics. But it does not mean that natural numbers "are" sets (or objects in a topos), any more than Quicksort "is" a piece of C++ code.
Further, the "modern" way to discuss this is not categorification, despite what Baez says; people who work and publish on the foundations of mathematics almost universally do so in the language of set theory.