In case any hunter had the same problem like me and so far is unable to make screenshots with any of the mentioned tips and workarounds, I now found a solution that works at least for my system. Fraps did not work for me. FSscreen did not work for me. Greenshot did not work for me. Windows PrintScreen key and saving the notepad content did noit work anyway.
But
MSI Afterburner did - and my nVidia board even is no MSI board.

Crystalclear, lasersharp full-size screenshots at gameplay resolution.
I think I will now run photosafaris and let the poor animals live - at least more often.