It seems you are missing some very basic knowledge, if you have questions like this. Watch/read some tutorials to get the basics, than ask specific questions.
This guy does the same thing as you: https://www.youtube.com/watch?v=yLduQiQXorc
This was like the 3rd result for searching for nginxproxymanager on yt.
You type the ip of the rpi on the router, so from an external call the router will forward it to the rpi. Or I don’t know what is your question.
Things may seem automagical in the networking scene, but you can config anything the way you want. Even in nginxproxymanager you can edit the underlying actual nginx configs with their full power. The automagic is just the default setting.
External 80 to internal 80 and external 443 to internal 443
With this config you don’t have to deal with ports later, as http is 80, https is 443 by default.
If you run some container on port 81, you have to deal with that in the reverse proxy, not in the router. E.g. redirect something.domian.tld to 192.168.0.103:81
If you use docker check out nginxproxymanager, it has a very beginner friendly admin webui. You shouldn’t forward the admin ui’s port, you need to access it only from your lan.
Very nice project! Thank you for using OpenStreetMap! I love it when the project I contribute to gets used in interesting projects like this!
But some quick notes, related to the map display: It’s call OpenStreetMap, there is no s at the end, written in CamelCase without spaces. The other more important problem is you forgot to include the attribution text on the map. For using OSM there is only one requirement, you have to display “© OpenStreetMap” somewhere on a corner of the map. More info about this on the website of the OSM foundation: https://osmfoundation.org/wiki/Licence/Attribution_Guidelines
I see the attribution text is displayed on http://trails.tchncs.de/ but not on https://demo.wanderer.to/ so I don’t know what’s going on.
The basemap display on the demo website uses the tile server from openstreeetmap.org. This is very discouraged, and also can give bad experience to users. The tiles on osm.org are raster tiles, they are regenerated automatically after a change in the map data, they are aimed as a tool for map contributors, not end users. You can read more about this here: https://operations.osmfoundation.org/policies/tiles/
There is a new totally free maplibre compatible vector tile provider, which uses the same map data, I recommend to switch to OpenFreeMap. Users can also self host OpenFreeMap, so some really privacy minded users could totally self host the full project this way.