Thank you @Raphael for the follow-up on this! I think I have found the reason although I can not understand the logic. But first lets set the stage…
I setup a new VPS, to make a side-by-side comparison with the problematic VPS. Lets call them VPS-PR the VPS with the PRoblem and VPS-OK the freshly installed VPS which is behaving as expected.
I understand that every Web domain is created by the bash command v-add-web-domain. So I had a closer look at line 128 of v-add-web-domain which lead me to line 43 of ip.sh, which in turn lead me to have a look inside /usr/local/hestia/data/ips/[IP HERE].
In there I found the difference in the two VPSes and then it hit me! Some time ago, while trying to solve the problem I described here, I set the option Assigned Domain (Optional) (Web UI > Server > Network > Edit IP) to the FQDN of the VPS-PR.
It seems that whatever is set in that box, will cause the creation of extra Domain Alias, when creating a domain.
For example, if the Assigned Domain (Optional) is set to TEST, then every new Web Domain that is created (e.g. example.com), is going to have an alias of [Web-Domain].TEST (e.g. example-com.TEST.)
The VPS-OK, didn’t have that option set. So every Web Domain I created, was fine. When I set the option Assigned Domain (Optional) in VPS-OK, then the same problem appeared. Now I can reproduce the problem 100% of the times and I believe you will be able too.
When I cleared the box Assigned Domain (Optional) in the VPS-PR, then new Web Domains are created without any extra alias. Which leads us to the question: What is the purpose of Assigned Domain (Optional) setting and when (if ever) should anyone set something in there?
I suppose we can mark this thread as solved, unless someone wants to shed some more light in this… 