using trailing slashes in nginx configuration

These locations are different. First one will match /production for example, that might be not what you expected. So I prefer to use locations with a trailing slash.

Also, note that:

If a location is defined by a prefix string that ends with the slash character, and requests are processed by one of proxy_pass, fastcgi_pass, uwsgi_pass, scgi_pass, or memcached_pass, then in response to a request with URI equal to this string, but without the trailing slash, a permanent redirect with the code 301 will be returned to the requested URI with the slash appended.

If you have something like:

location /product/ {
    proxy_pass http://backend;
}

and go to http://example.com/product, nginx will automatically redirect you to http://example.com/product/.

Even if you don't use one of these directives above, you could always do the redirect manually:

location = /product {
    rewrite ^ /product/ permanent;
}

or, if you don't want redirect you could use:

location = /product {
    proxy_pass http://backend;
}

No, these are not the same--you will need to use a trailing slash with a regex to match both, i.e.

location ~ /product/?

See this related answer for a more detailed response on how to match the entire URL.