Terms like "agreeable", "socially mature", and "well balanced" (versus "attention seeking", presumably a bad thing) can signify that parents place quite a bit of importance on how well their children conform to social norms and expectations. The article mentions bedtime rules, but this can also extend to long-entrenched traditions in "Old World" societies.
On the other hand, the fact that American parents don't use these terms, and instead use terms like "independent" and "rebellious", shows that American culture places a high value on thinking and living outside of the box. The lack of "calm" and "happy" might make life more stressful for parents and children alike, but at the same time, the term "adaptable" implies that Americans expect the world around them to change quickly, and when it does, you gotta change, too. America is newer and changes faster. It's not surprising that Americans value qualities that make people innovative, disruptive, and not bound by any rule that they didn't make by themselves.
As a person who has lived half his life in a society where being "well rounded", conformity, and obedience to authority are considered a child's primary virtues, I must confess that I am rather partial to the "American" style, tradition be damned.
Edit: There also seems to be a relevant difference between Northern/Western Europe and Southern Europe, just as there is a large cultural difference between Northeastern/Western U.S. and the Southeast. So maybe the comparison should be between industrial regions and traditionally agricultural regions, or between relatively secular regions and deeply religions regions, rather than between America and Europe as a whole. Seems obvious in some ways.