Website using wrong URL after proxing though Nginx

This is my first time using nginx and I'm finding this situation quite puzzling. My website is hosted at web.example.com and in order to reach the backend, requests are routed though nginx as a reverse proxy. After processing through nginx, the URL changes to the URL of the nginx server (proxy.example.com) instead of being the URL of the website.

The flow looks something like this:

web.example.com -> nginx (proxy.example.com) -> backend (service.example.com)

When I log the request in the backend service, I can see that the x-forwarded-host is set to proxy.example.com. Is this correct or should it be set to web.example.com here?

The nginx.conf file is:

events {
    worker_connections  1024;
}

http {
    include    /etc/nginx/mime.types;
    default_type application/octet-stream;

    log_format   main '$remote_addr - $remote_user [$time_local]  $status '
        '"$request" $body_bytes_sent "$http_referer" '
        '"$http_user_agent" "$http_x_forwarded_for"';

    access_log   logs/access.log  main;

    server { # simple reverse-proxy
        listen       80;
        server_name  proxy.example.com;

        location /site {
            proxy_set_header Host $http_host;
            proxy_set_header x-forwarded-for $proxy_add_x_forwarded_for;
            proxy_set_header x-forwarded-host $host;
            proxy_set_header x-forwarded-server $host;
            proxy_set_header x-real-ip $remote_addr;
            proxy_pass_request_headers on;
            proxy_pass http://service.example.com$request_uri;
        }
    }
}

How do I get the URL to go back to web.example.com/site after going through this flow instead of showing as proxy.example.com/site?

Thanks.