SchulzStefan
I agree with Stefano's advice, but that's not the only way or answer.
In the past (until today) I set up all my servers for the primary domain always with something like "abc.local". Further then, I created one or more virtual domains with the name of real existing internet domains, like "xyz.de".
Well that "works" but you just encountered one of the reasons why using abc.local for the main domain name is not a good idea.
As I said before, far better to use a resolvable domain name, the server does pass those credentials to other servers & mail will not be rejected in that case.
...is my configuration a mis-configuration? Would it be better to change it? And if so, is it just enough to rename the domain in the admin-panel? Or is it better to follow the rule - never touch a running system?
It is generally safe to make changes to a SME server, providing tweaks have been done using custom templates or db commands, as these changes will be retained during a system reconfiguration & reboot. The main domain can safely be changed, a new self signed certificate will be created for the new domain name & other data & settings will be retained, including the virtual domain names.
Not necessarily a mis-configuration, but not necessarily "best practice".
AFAIK for the last 14 years, SME server developer advice has always been to use a resolvable domain name, & it can be the same domain name as your main site production server, particularly if the secondary server is behind that. It depends on your network & requirements, but there have been many problems reported in these forums over the years to do with servers having a main domain like abc.local.
The secondary servers server name should be different so each server can be identified.
I tend to use one of the hosted domains as the main domain name, & all other hosted domains are virtual domains, added in the server manager Domains panel, or create a free (& resolvable) domain name as the main domain (which is not really used by anyone), & then add other "active" domains as virtual domains. It depends on your specific needs & server & network configuration etc.