This is a complex issue it seems.
I get no rain in PP. I do see a second of rain in VLC, then clean. When I open the clips in my default media player, KMPlayer, I have the option of selecting the renderer I'd like to use. Some renderers (like the default overlay) show little or no rain. Others (like the Direct3D 9 RGB Surface) show it very clearly, worse than I've ever seen on any other clip. So it now seems as if the renderer plays a part in all this. Why these clips show rain in VLC, but not PP, I haven't a clue. (But I am grateful as hell, not having to deal with this on a production basis.)
I still think the error lies at least partially in the clips themselves. Something is causing them to be rendered improperly, where AVCHD from other cameras render fine. (At least, the GH2 is the only camera I've heard of having this issue.)
So now, I don't even know what question to ask next. Your clips show the rain in PP, mine do not. Yet they both show rain in VLC and other players. What the HELL is going on here?! There's no consistency to grab hold of, no one factor that remains the same in all cases.
(Mine do have a GOP structure, by the way. They're recorded using the Cluster v7 'Apocalypse Now' 12/15 GOP (DREWnet) with cbrandin '444 Soft Matrix'.)