You need several pieces to make this work. Thinking through the steps of how a request could be handled:
DNS for www.mywebsite.com points to a single IP, there's no way around that. So all requests for that host, no matter the protocol, will come in to the machine with that IP, the public server.
So we need to route those requests, such that a) https requests are handled by nginx on that same machine (the public server), and b) http requests are forwarded to your local machine. nginx can do a) of course, that's a normal config, and nginx can also do b), as a reverse proxy.
Now the problem is how to route traffic from the public server to your local machine, which is probably at home behind a dynamic IP and a router doing NAT. There are services to do this but to use your own domain is usually a paid feature (eg check out ngrok, I guess Traefik probably handles this too, not sure). To do it yourself you can use a reverse SSH tunnel.
To be clear, this routes any request for http://www.mywebsite.com/ to your local machine, not just your own requests. Everyone who visits the http version of that site will end up hitting your local machine, at least while the tunnel is up.
For 1, you just need your DNS set up normally, with a single DNS record for www.mywebsite.com. You don't need any /etc/hosts tricks, remove those (and maybe reboot, to make sure they're not cached and complicating things).
For 2, your nginx config on the public server would look something like this:
# First the http server, which will route requests to your local machine
server {
listen 80;
server_name www.mywebsite.com;
location / {
# Route all http requests to port 8080 on this same server (the
# public server), which we will forward back to your localhost
proxy_pass http://127.0.0.1:8080;
}
}
# Now the https server, handled by this, public server
server {
listen 443 ssl;
server_name www.mywebsite.com;
# SSL config stuff ...
# Normal nginx config ...
root /var/www/html
location / {
# ... etc, your site config
}
}
The nginx config on your local machine should just be a normal http server listening on port 8080 (the port you mentioned it is running on). No proxying, nothing special here.
For 3), lastly, we need to open a tunnel from your local machine to the public server. If you are on Linux, or macOS, you can do that from the command line with something like this:
ssh user@www.mywebsite.com -nNT -R :8080:localhost:8080 &
If you're on Windows you could use something like PuTTY or the built in SSH client on Win 10.
The important parts of this are (copied from the SSH manpage):
-N Do not execute a remote command. This is useful for just forwarding ports.
-R Specifies that connections to the given TCP port or Unix socket on the remote
(server) host are to be forwarded to the local side.
The -R part specifies that connections to remote port 8080 (where nginx is routing http requests) should be forwarded to localhost port 8080 (your local machine). The ports can be anything of course, eg if you wanted to use port 5050 on your public server and port 80 on your local machine, it would instead look like -R :5050:localhost:80.
Of course the tunnel will fail if your public IP address (on your localhost side) changes, or if you reboot, or your local wifi goes down, etc etc ...
NOTE: you should also be aware that you really are opening your local machine up to the public internet, so will be subject to all the same security risks that any server on the public internet faces, like various scripts probing for vulnerabilities etc. Whenever I use reverse tunnels like this I tend to leave them up only while developing, and shut them down immediately when done (and of course the site will not work when the tunnel is down).