Ok, fixed that now - I didn't have a root user record on the db-server. The command "mysqldump -h db1.domain.tld -u root -p c1testdb1 > c1testdb.dump" works correctly now when I execute it on my ISPConfig master server.
However, the DB-Backup still doesn't work. The ISP-Config interface shows the backup, with the "restore" and "download" options, but there's no db-backup in the /var/backup/.. folder, and neither "restore, nor "download" does anything.
The password in my mysql_clientdb.conf file definitely is correct. What else could be the problem?