I hae set up several test setup's. Onced I used a salve domain which was dedicated to a full url and I have added the standard joomla template for it.

Ok, now I have deleted the old site and set up a new site. Then I have enter the domain name for this site as well. but what happen now:

http://{site_url}/sites/u{user_id}s{site_id} - show the right template I have selected for the site

my-domain.com - shows up the right content, but with the wrong template (joomla standard) with the domain I have tested before

I can not find any way to get around this. Is it some where wrtieen in a table? if yes, in wich?

I don' understand your question.

When you create a website from the front-end, it consists in the replication of an existing website with all its content and setup.
So you will have the same menu, same default template selected, ...
So start to check the website that are using for the replication and check its setup before trying to replicate it.