Oops, I just realized this isn't quite true (though it very much felt like it for a newbie). If I turn off "Reselect active region", then the auto-select settings works a little closer to "as expected".
I say a little closer because, if "auto select active window" is on and "reselect active region" is off, the behavior I see is that the screenshot comes up with the active window selected, but then as soon as it finishes saving the image, it de-selects the active region.
It seems that the "reselect active region" setting takes precedence over the "auto-select active window" setting in all cases, which makes it appear like the auto-select-active-window setting is broken.