Hi Micha,
I was just about to reply for the benefit of anyone else following this discussion when I looked again at your post and saw the last photo with the extra seal in it. Most standard caps look like the next photos and wont seal the shiny disc to the filler neck.
It took me a while many years ago to work out where the air was getting in until I found this little valve.
My first car (triumph herald) had exactly the arrangement you have pictured above with a plastic bottle tied to the front of the radiator but never worked properly. One of those double seal caps would have corrected it if I had realised where the air was getting in. Who supplies these?
Andrew