Maybe I just found something...
For any reason that I do not get so far, in the class
SolrNetContainerConfiguration
in the method
AddCoresFromConfig
Each and every core is registered again, every time a new container is set up.
So I set up my container with the new connection for logging,
and afterwards I add another core with the "normal" connection,
which then always overwrites the old ones.
The really weird part for me at present is, that I had already set up my own mapping collection,
in which I put an own SolrServers instance for each document type I have and each of these has its own unity container instance;
which I did exactly with the intention to prevent anything like that.
So I really do not have a clue, how this possibly could have overwritten my registrations in the other containers,
but after I put the registration of the other cores BEFORE the updated of SolrConnection, it suddenly worked...