homeassistant

HA doesn't need either of these, but if you want an SSL certificate (to run over HTTPS instead of plain HTTP) it is bound to a domain name, which must be public unless you want to enter in the zone of adding our custom certification authority to each of your devices. This name is resolved by a public DNS. You asked how to use it when internet is down, in this case a public DNS is not reachable so you need your own on the local network.

The reverse proxy is useful when you have a bunch of web services and you want to protect all of them with HTTPS. Instead of delivering the certificate to each of them, you add the HTTPS layer at your reverse proxy and it queries the servers behind in plain HTTP. The reverse proxy has also the benefit of making handling subdomains easier. So instead of distinguishing the different services because they have a different port number you can have a few https://ha.my.domain/ and https://feedreader.my.domain/

If you just have homeassistant and not care of HTTPS the easiest option is to use the local resolution: modern OSes advertise the name of the device on the network and it can be resolved on the .local domain. But, if you configured HTTPS to use https://name.duckdns.org/ you'll se an error when you try to use https://name.local/ because your browser sees a mismatch between the name in the certificate and the name that you are trying to connect to. You can always ignore this error and move on, but it mostly defeats the point of HTTPS.