Equivalence of Definitions of Topology

Definition 1 implies Definition 2
Let $\tau$ be a topology on $S$ by definition 1.

$O 1$ is the same as $O 1'$, so $O 1'$ holds for $\tau$.

From General Intersection Property of Topological Space:
 * if $O 2$ holds, then $O 2'$ holds.

Thus $\tau$ is a topology on $S$ by definition 2.

Definition 2 implies Definition 1
Let $\tau$ be a topology on $S$ by definition 2.

$O 1'$ is the same as $O 1$, so $O 1$ holds for $\tau$.

$O 2'$ states that the intersection of any finite subset of $\tau$ is an element of $\tau$.

This applies when the subset of $\tau$ contains exactly $2$ sets.

Thus $O 2$ is a direct consequence of $O 2'$.

Also as a consequence of $O 2$, it follows that the intersection of an empty subset of $\tau$ is an element of $\tau$.

From Intersection of Empty Set it follows that $S \in \tau$.

So $O 3$ is a direct consequence of $O 2'$.

Thus all the open set axioms hold.

Thus $\tau$ is a topology on $S$ by definition 1.