Thanks Francisco and Antonia for pointing out the problem and the solution!
The newer newest version of Collect Earth (1.3.9 from today 5th of July 2016) addresses this problem! So that even the KMLs generated directly from ArcGIS/QGis should be working fine.
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On another note, I have also noticed that sometimes when downloading a CSV file from a **Fusion** table some of the **polygon descriptions are "cut"** so that part of the KML Polygon element is missing ( the last few hundred characters).
This might happen with very complex polygons! So beware of this issue! When you download the CSV from Fusion tables make sure that the polygon descriptions end with </Polygon>