Thanks! I stand corrected. The timing of acceleration also matters. I should have known better. Anyway, I agree that

The reason the astronauts age less is that the path they follow through space-time corresponds to a smaller proper time than the path followed by people who remain on the Earth, and the proper time along a path is what a clock following that path measures.

It just seems like a tautology to me (the difference in aging is due to the difference in subjective clocks). To cause this difference one has to make the worldlines diverge, and this means difference in acceleration profiles.

What I initially was unhappy about is the statement

You're right that Cowen got it backwards, but you're wrong about this:

Acceleration is what causes the opposite, e.g. turning the spaceship around to come back

That last statement is perfectly correct.