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

Acceleration is not the cause. 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. So it's a geometrical fact about the difference between the two paths that causes the asymmetrical aging, not the acceleration of the astronauts.

To make this obvious, it is possible to set up a scenario where another group of astronauts leaves Earth and then returns, accelerating the exact same amount as the first group, but following a path with larger proper time. This second group of astronauts will age more than the first group, even though the accelerations involved were the same.

A lot of elementary presentations of relativity identify acceleration as the relevant factor in twin paradox type cases, but this is wrong (or, more charitably, not entirely right).