The proxy server (if you were accessing it from your webserver) may only route traffic that it can access, otherwise may simply just try to show you the page if not. I'm sort of baffled on that one too.
I'm certainly not a network guru, but there are a few more steps you can take to identify where your having an issue.
First off, try assigning the webserver a static IP on your internal network. Then try accessing it from another computer (assuming you have another) on that network using the IP. If you are successful, that means there is no issue with the server hardware itself or software installed on the server. If you can't, check you network card and network config and ensure your physical connection from the router to server (which if you can access the internet shouldn't be a problem). If you're able to ping the internal address, but not access your website, then it is likely a firewall/apache configuration issue.
Check the manual that came with your home router to make sure port forwarding (port redirecting) is an option. What you will need to is forward the port that is handling internet traffic (8888) to your server IP. Again, this is highly dependent on your router configuration. You will likely need to assign the IP that Cox gave you to the router itself. I understand that Cox may occasionally change these IP's so be wary of that. Make sure the gateway address in your servers network configuration is the internal IP of the router. And again, make sure you have a static IP in your router for your server.
Make sure that when you are trying to access your server at all times you use http://xxx.xxx.xxx.xxx:8888
, no matter if you're on an internal or external network. Because connections are made on port 80 by default, just typing the IP address will send requests over port 80 and will miss 8888 altogether.
Sorry for the book. Its just that for beginners, setting up this kind of network requires a lot of patience and a fairly high learning curve and is different for just about everyone depending on their equipment and circumstances. It requires a slow, step by step, thorough setup.
Again, Good Luck.