Well ... no humor me ... do info on df -h ... so let's focus on what it says ..
"says that the file in the local data directory could not be created and that I should check the rights"
Example: destination is /home/coursebackups/ Do both directories exist? Moodle script won't create them for you ... they must exist already.
What are ownerships and permissions on destination:
ls -lR /home/coursebackups
"backup.mbz got created in the temp directory"
That's the file it is trying to copy ... the size is kinda large but server should be able to handle it ...
What if you, changed into the tmp directory and issued a copy command yourself to the desired destination.
cd moodledata/temp/backup/aslwerjweasdfasdf (where backup.mbz file resides)
cp backup.mbz /home/coursebackups/
Do that error also?
What if you used sudo for the user web service is running under:
sudo -u webserviceuser cp backup.mbz /home/coursebackups/