You can see the piece of glass at 1:57 in the dailybeast video: http://tinyurl.com/8x5cp5aThe Daily Beast article states there is a shard of glass lying on the window sill. No doubt this is the same piece of glass Fleet White said he picked up and placed up there.
Now, go to 1:30 in the video. Here you can see that when the window is closed it is flush with the trim that runs around it. This means that if the window was closed, or even only slightly ajar there would be no ledge for White to use.
Also, glass on the ledge should not be able to move from one side to the other unless someone picks it up and moves it because there is a lip, or ridge that divides the inner and outer window ledge.