Welcome to the community, @vanhims. Thanks for describing your domain configuration in great detail and offering your real domain and IP address for diagnosis, it enabled me to take a quick look at your DNS records and helped me understand your situation better. I don’t know why people attempt to obfuscate their domain names and IP addresses when seeking help here in 2020 when privacy is dead anyway.
Well, before I address the primary issue of getting 3designs.be resolve to the Virtualmin box, I would like to point out that Virtualmin’s web interface can be accessed from any domain / virtual server that is hosted on the Virtualmin box, thus: virtualserver.com:10000 or (if admin subdomain has been pointed to Virtualmin box’s IP) admin.virtualserver.com, which makes it much easier to type for lay people. You do not need to enter your IP address 188.166.83.142 in the browser (and bypass the security warning) every time you wish to access Virtualmin, @vanhims; just use the domain name of any of the virtual servers you host. 
Now, about the matter of getting 3designs.be resolve to the Virtualmin box:
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3designs.be will not direct to ns1.3designsbe. 3designs.be and www.3designs.be will point to 188.166.83.142. ns1.3designs.be will point to 188.166.83.142. ns2.3designs.be will point to 188.166.83.142.
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You will then configure Virtualmin to use ns1 and ns2.3designs.be as name servers for current virtual servers + sub servers and aliases as well as for any that you create in future. This is done via https://www.virtualmin.com/documentation/dns/faq#How_do_I_setup_nameservers_for_my_server
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Finally you could create admin.3designs.be and point it to 188.166.83.142 so that you would be able to access Virtualmin’s web interface by entering admin.3designs.be in the browser.
If all goes well, you will have 3designs.be resolving to its assigned virtual server configuration in Virtualmin and be able to use your vanity nameservers ns1 and ns2.3designs.be on current and future domains which you host on Virtualmin.
As a bonus, you will be able to admin Virtualmin via admin.3designs.be instead of having to use the IP address.