Quasi-separatedness as a topological condition on the scheme

Yes. A reference is EGA IV, première partie, Proposition (1.2.7), b'), page 228.

Numdam link

Another reference is Tag 01KO in the Stacks project (note that when $S$ is affine the hypothesis that the two opens map into a common affine is empty).

Of course, the condition that the intersection of any two affine is quasi-compact, is equivalent to the condition that the intersection of any two quasi-compact open subsets is quasi-compact.