Looking at the example in the article, I'd say that the one on the LEFT is flat. The one on the right is just completely borked.
Have a look. On the left, the "shop" buttons are blue, caps, underlined. No 3d shading or even 2d box around the border of the button. That, to me, is "flat". On the right, the "shop" buttons are in the same style as the body text - grey, mixed case. You could only find them by guessing from the words that it is clickable, or by trial and error (unless there is a mouseover).
This does not appear to be an argument in favour of the pre-iOS 7, and equivalent Windows, button styles. It's just an argument for not completely borking your UI.
It's worth looking at all their tests. Some are better than others:
Approximately the bottom third of the page is tests where they didn't find any difference between the two versions.