Hi there!
Thank you very much vor your help.
Indeed when I list all listening ports on the server there is no fastcgi listening on Port 9000. One look at phpinfo.php shows that PHP-FPM/fcgi is active/enabled and used, but not over any port. As @besendorf said, this is most likely handled by nginx.
I replaced then the fastcgi block in the caddy file with the line proxy / nginx:8080 as @besendorf (scheint wohl ein ein ziemlich sauberes Dorf zu sein! 
) suggested but I have the same issue with bad gateway.
I looked then in /var/log/nextcloud_errors.log for any errors and this is what I get:
root@DietPi4:~# cat /var/log/nextcloud_errors.log
2020/01/31 11:20:46 [ERROR 502 /index.php/] dial tcp: lookup nginx on 192.168.5.1:53: no such host
2020/01/31 11:20:46 [ERROR 502 /index.php/favicon.ico] dial tcp: lookup nginx on 192.168.5.1:53: no such host
2020/01/31 11:22:53 [ERROR 502 /index.php/] dial tcp: lookup nginx on 192.168.5.1:53: no such host
2020/01/31 11:22:53 [ERROR 502 /index.php/favicon.ico] dial tcp: lookup nginx on 192.168.5.1:53: no such host
.
So it seems that caddy is trying to reach nginx on Port 53 on my Gateway/Router instead on the server itself.
There is no line in the Caddyfile where I have put the IP of my Router/Gateway in. So it seems that caddy is trying to reach the nginx Server on the DNS-Server Port of my Router. That is some odd behavior. Or did I miss here something?!
Have a nice weekend and thank you very much for your help!