Definitions of $2$ in set theory?

The first definition is a good practical example of a set that has two elements. See it as the "standard" example of what a set with two elements looks like, a representative of the concept of $2$.

The second class you describe is the collection of all things that have two elements, so it completely describes the concept of having the quantity $2$. The problem is that the second class is not a set: it has too many elements, and is therefore a proper class. This makes doing mathematics with it a little troublesome.

What the second class describes, is the concept of cardinality: it describes all the sets that have cardinality $2$. Another way to define this class, would be to take all the sets that have a bijective function to the representative set $\{\varnothing,\{\varnothing\}\}$.

Your first example (Von Neumann's definition) is the most typical set-theoretical definition of $2.$ There are many other choices that could be made, but this one has advantages, most prominently that it generalizes to the usual nice definition of ordinals.

Your second example (I think there are typos but what you intend is clear) is not a set at all in the standard framework (ZF), but rather a proper class. This is the class of all sets with cardinality two (or equivalently with the "twoness" property). Your first example could be said to be a canonical representative of this class.

However, it is perfectly reasonable to have a definition of two that does not have the twoness property. For instance, the Zermelo definition has $0=\emptyset,$ $1= \{\emptyset\},$ $2=\{\{\emptyset\}\},$ etc. What's important is that there is some way of mapping the set back to the intuitive idea of the number two in the context of the definitions of the rest of the natural numbers, not that it literally has two elements. Though one might view the fact that the cardinality of a natural number is the natural number itself as another advantage of the Von-Neumann definition.

To answer your question directly, don't try to think that one of these definitions is "the correct one". Each of these definitions is a different means of establishing 'twoness', each serves a different mathematical and/or philosophical purpose, and each is "correct" in its proper context. I like to think of them as different 'standard rulers' for measuring 'twoness'. [personally, I think the second definition is closer to 'correct' for general 'twoness', but unfortunately, as Russell pointed out, the underlying mathematics doesn't actually work right]