Update:
I e-mailed the guys at the webpage. They were very friendly and told me the following:
The “superflip” they show has a 22-move process because they are using a java app to display the way to get to a certain position, and the java app calculated a 22-move solution. The page clearly states that it should be possible in 20 moves.
It is standard in the cubing community to count a 180° turn is one move instead of two. If you count 180° turns as two moves (which I thought you should do), then the “God’s number” (minimum number of moves to solve a worst-case scenario) would be somewhere between 26 and 29 moves.
From the e-mail, discussing the case where 180° turns count as two moves: