Safari for one user can't connect to local server via https due to new cert
Seems I finally found the solution (after trying so messing so many things up due to other solutions):
Using "Internet Accounts" to add a CalDAV account for the server.

As soon as I tried to add a new CalDAV account for "https://old.mynas.local" I got a "untrusted certificate - what should I do"-dialog and could successfully trust the new certificate. Then all problems in Safari vanished as well.
It seems OSX internet accounts uses its very own certificate system and this trusted one of the old certificates due to a CalDAV account previously configured and thus forbid access to the https server now serving the "wrong" certificate. The creepy thing: the dialog popping up was in English although my OS language is not English. 

This solution might even work if one doesn't use CalDAV, because the certificate dialog should pop up as soon as the https connection is established, even before trying to check for DAV functionality.