The last video in the backyard - I would like to see you take the same one with the GH2 - you will see quite a bit more detail in the trees and leaves and even the grass.
The Sony appears cleaner because of the noise reduction, but there is definitely detail that is squashed to get to that point. If you like to pan a lot in your videos handheld, you obviously would not want to use the 1080p24 mode of the GH2 (or pan slower).
Hey, I sold my GH2 - I preferred the look of the Sonys as well. I love slowing down 1080p60 to 1080p24 in post for slow motion (.40 playback). Looks much better than using Twixtor. But, all things being equal, if you take a GH2 and put it on a tripod and pan slowly it will out-resolve the NEX/Alpha cameras by quite a bit detail-wise.
But I shoot my kids - so I want 60p and better colors and IS, so I went with Sony.