Nginx, rewrite or proxy_pass. How to translate from site.ru/page/ to page.site.ru?

Question

Nginx

Good day!
nginx + php-fpm config.
Website site.ru
How to transfer the internal page page to a subdomain? For example:
site.ru/page/ -> page.site.ru ?
It is necessary that the page opens on both URLs, but in the address bar, so that it is page.site.ru

Answer 1

You make two servers (two virtual hosts). First:

server {
    server_name  site.ru;
    location =/page/ {
      rewrite ^(.*) http://page.site.ru$1 permanent;
    }
}

and the second one with a subdomain:

server {
    server_name page.site.ru;
   ...
}

This is a simple but not the best solution.

Answer 2

I checked the configs above, it turns out like this:
the 1st host, when trying to go to site.ru/page, redirects to page.site.ru/page/, but there is no such page, so error 404
2nd host what rules to set?
If you change the 1st config, remove $1, and specify proxy_pass site.ru in the second config, then when you go to site.ru/page, it will redirect to page.site.ru - which was required, however, the content will be the same as on the main page (site.ru), not page.
1st host

server {
  server_name site.ru

  location =/page/ {
            rewrite ^(.*) http://page.site.ru permanent;
        }
}

2nd host

server {
  server_name page.site.ru

  location / {
    proxy_pass  http://site.ru/;
  }
}

If you put proxy_pass site.ru/page in the 2nd config, then as a result there will be errors with cyclic redirection.
How to be?