The most commonly used or the most standard version of the axioms in ZFC and the axiom of pairing?

Question

[Question 1]
What are the most commonly used or most standard axioms of ZFC written in logic (and set theoretic) language?

[Question 2]
There currently seems to be 2 different version of the axiom of pairing in ZFC on the Wikipedia.
On https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#4._Axiom_of_pairing it is:
$$(\forall A)(\forall B)(\exists C)((A \in C)\wedge(B \in C))$$ On https://en.wikipedia.org/wiki/Axiom_of_pairing it is: $$(\forall A)(\forall B)(\exists C)((\forall D)(D \in C \iff (D=A \vee D=B)))$$ The 2nd one implies the 1st one. Which one is the most commonly used or the most standard?

Answer 1

It's hard to say "standard". People don't normally care about the exact axioms used, just the consequences of these axioms.

For example, once you've included the Axiom [schema] of Separation, you can weaken a lot of the other axioms to be like your first version of Pairing.

But Separation follows from Replacement, assuming we phrase Replacement in a "strong way". So some people will omit it entirely.

Similarly with Pairing, the axiom is in fact a consequence of Replacement and Power Set. So it can be omitted, or it can be included.

The exact choice of axioms depends on your use of them. If you want to study weak set theories, adding axioms like Pairing and Separation is important, similarly this will influence the answer of which way you want to formulate your axioms. But if you want to only study consequences of $\sf ZFC$, then nobody cares if you formulated a particular axiom this way or that way. Much like nobody cares if you include Pairing in your list of axioms of not.