When I type https://homebridge.mydomain.com with a port I get a 502 Bad Gateway

This is an issue with your reverse proxy configuration and unrelated to the "hostname" config option.

Setting the host name is not required if you correctly configure nginx. An example config can be found here:

https://github.com/oznu/homebridge-config-ui-x/wiki/Reverse-Proxy:-Nginx-and-SSL

I am not looking for a way to install the SSL certificate into the homebridge. Thats why I have the the Reverse Proxy. The homebridge should just respond to simple http trafic but for some reason it does not.

The location block in the example is relevant for both http and https endpoints.

location / {
    proxy_pass                  http://127.0.0.1:8080; # replace 8080 with the port the ui is running on
    proxy_http_version          1.1;
    proxy_buffering             off;
    proxy_set_header            Host $host;
    proxy_set_header            Upgrade $http_upgrade;
    proxy_set_header            Connection "Upgrade";
    proxy_set_header            X-Real-IP $remote_addr;
    proxy_set_header            X-Forward-For $proxy_add_x_forwarded_for;
  }

I am not getting what you are saying at all I am afraid.

Are you telling me to update the homebridge setup or the Reverse Proxy ?
And if you are saying the Reverse Proxy I need to have some more understanding of why the homebridge does not work out of the box as all my other images that pass through the Reverse Proxy?

Your reverse proxy is not working. A 502 means the connection from your reverse proxy to homebridge was never established.

A reverse proxy running nginx requires those config options linked above to work fully with Homebridge as these are needed to proxy web socket connections.

However since you're getting a 502 - this probably means there is another issue you need to solve with your proxy container first.