Exactly, as you say it looks like Contabo is routing the request through its own systems.
Can you please open a support ticket with your web host, they might be able to help with the issue.
Looks like that website is using XenForo, but this Nameless template looks similar: https://namelessmc.com/resources/resource/142-modernx--premium-template/
To get more information about this error, you can edit the file core/classes/Email.php in a text editor. Delete all code inside, and replace with this new code.
Then, when you send a test email in the StaffCP, a full debug log will appear on screen with the exact communications between your web server and the SMTP server. This should include some hints as to what the problem is.
Once you have the full log, revert the contents of core/classes/Email.php back to its original code.
Usually this means there is a firewall blocking the connection to the email server, or the SMTP address/port combination is incorrect.
You can also check the credentials in the file core/email.php, and if secure is set to tls you can try changing it to ssl and vice versa.
Can you please make sure the resources module has been correctly uploaded, and that the folder within the modules folder is named Resources and not resources