Nginx – Convert a RewriteRule from nginx to apache

It appears as though you have a routing loop.

You are directing all traffic from / to /public and then all traffic from /public to /index.php

Based on /public/.htaccess, all requests to /public should be routed to /public/index.php?url=$1 instead of /index.php?url=$1

 location /public {
        if (!-e $request_filename){ 
          rewrite ^(.*)$ /public/index.php?url=$1 break;
        }
      }

First, you don't need the location ~* ^/(css|img|js|flv|swf|download)/(.+)$ block, because it doesn't change anything. Also, in general, root directory inside location blocks isn't what you usually want, alias is the thing you most likely want.

But to the actual problem, try adding these location blocks:

location /v1/documentation/ {
    rewrite ^/v1/documentation/(.+)$ /v1/public/documentation/$1 last;
}

location /v1/examples/ {
    rewrite ^/v1/examples/(.+)$ /v1/public/examples/$1 last;
}

An optional optimization for your @rewrite block

location @rewrite {
    rewrite ^ /v1/public/index.php?_url=$uri break;
}