This is all very well known to me. The problem is that the Apple TV view on the iPad was different from my other devices, and the Rented Movies tab would not show at all, as I explained in my question.
I suspect the reason was a corrupt backup file, the corruption originating from the days when I was using iTunes was syncing my iPad (incl. movies) with my computer. I reseted the iPad back to factory setings and started using it as new without downloading any backups. Now the Apple TV app looks as it should, and the rented movies show correctly.
However, there is a NEW PROBLEM. I cannot download the rented movies, only ones that I have bought. Downloading the rented movies proceeds normally to the end, but then there is an error message: the movie cannot be downloaded. This has happened repeatedly with different movies. This is not an issue with the internet, since I have succesfully downloaded movies that I have purchased. The movies are not downloaded elsewhere, so that is not the issue either. (However, I tried downloading one on my computer and that worked. So the problem is with the iPad.)