How do I show a 403 error page when the server returns a 403 error code?
What you need to do is to configure nginx to serve a drupal page that generates a 403 page. There are severals ways to do this.

The easy solution is to route the traffic to Drupal (the index.php file), which would generate a 404 since the page doesn't exist in Drupal.

I think doing

location ~* \.(txt|log)$ {
  try_files @rewrite
}


should accomplish this for .txt and .log files. It should try to serve the file from Drupal. By default this will be a 404, but you can add the a route for specific files and return 403 instead.