Hi @Olaf Helper,
When you rename a Windows server name, IIS is already installed and most IIS functionality will continue to work without any intervention, but IIS will not automatically update to the new name.
As you said, you can rename the server name in IIS by right clicking on the server name in the left connection pane of IIS. Choose Rename, provide a new name, and save the connection.
Not sure if this is the only way, but for IIS on windows server, under the Connections pane in IIS, right click on the server name under Start Page, select Remove Connection. Then right click on Start Page and select "Connect to Server...". Now enter a new name for the server click next and then finish.
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Best regards,
Yurong Dai