The situation is rather simple at a higher level of abstraction. We are interested here in a collection of objects U, equipped with some binary relation $\prec$, from which we can define two equivalence relations on U:
- $b_0\simeq b_1$ iff $\forall a, a \prec b_0 \iff a \prec b_1$;
- $a_0\backsimeq a_1$ iff $\forall b, a_0 \prec b \iff a_1 \prec b$.
An equality on U must be an equivalence relation $=$ for which all formulae all stable under substitution of equals by equals. In general, we would require that all operations preserve the equality and all relations are invariant under it. Here, we just need $\prec$ to be invariant under $=$, i.e.:
- $b_0 = b_1 \Longrightarrow b_0 \simeq b_1$;
- $a_0 = a_1 \Longrightarrow a_0 \backsimeq a_1$.
These are the left-to-right directions of your conditions 1 and 2.
Now, it’s obvious that doing the following is equivalent:
- Start from $U$, $\prec$ and an equality $=$ (satisfying 1 and 2), and require that $=$ coincides with $\simeq$ (i.e. the reverse of 1 holds).
- Start from $U$, $\prec$, define $=$ to be $\simeq$ (so that 1 and its reverse hold) and, to ensure that it is a proper equality, require condition 2.
Most often, people do the first and call the reverse of 1 (or the equivalence) the Axiom of Extensionality; some (like Takeuti and Zaring in Introduction to Axiomatic Set Theory, pp. 7-8 or Schechter in Handbook of Analysis and its Foundations, p. 29) do the second and call 2 the Axiom of Extensionality.
As Eric Wofsey mentions, the reverse of 2 holds if singletons exist, i.e. if for any $a$, there is a $b$ such that $x \prec b$ iff $x = a$: we apply the definition of $a_0 \backsimeq a_1$ to that singleton $b$ obtained from $a_0$ (or $a_1$). If we add everywhere the assumption of the existence of singletons, the two approaches above are thus also equivalent to
- Start from $U$, $\prec$, define $=$ to be $\backsimeq$ (so that 2 and its reverse hold), and require 1 and its reverse.