Finally I found a solution :)
Cinema is the problem in this case: it doesn't export normals in obj-files.
It fooled me, because when you open an obj-file in Cinema that doesn't
contains normals, it seems to generate them on the fly, so normals are
visible in Cinema, even if there were none in the file.
So for everyone whos interested in exporting working obj-files from Cinema
to broomstick:
You have to install two free plugins in Cinema - one for generating a normal
tag ("Add normals": http://skinprops.com/download.php?list.12) and one for
exporting proper obj-files (http://skinprops.com/riptide.php).
Just add the normals-tag to your object and then export it using riptide. In
the export options of riptide I set the scale factor to "1" and only
checked: "Export Normals", "Export Faces", "Reverse Faces" and "Export UV
Coords".
That was a hard bit of work ;). It doesn't explain, why the older broomstick
version could render my objects without normals properly, but who cares *g*.
Thanks Fabrice and John for your great help :)