How to setup 2 URLs pointing to same location in Apache for Rails

It does look like they're proxying. You can verify this by hitting the site and looking at the source IP -- if it's yours, then they're not proxying, they're doing something really weird, otherwise the source IP should be that of their proxy box. I would imagine that their proxying is probably doing all sorts of unpleasant things to the setup -- if the Host header being sent by their proxy server isn't what you're expecting, for instance, then that'll naturally cause problems. I always log the provided vhost name in my "default vhost" logs for this very reason -- it shows me what people might be hitting me from.

When it comes to the setup, I'd strongly recommend getting them to change the DNS to point to your server directly. That you're "leasing" the domain is irrelevant; DNS can still be managed properly. The only time proxying makes sense is if they're only redirecting part of the site to you, and serving other content on the same domain locally. For your Apache configuration, why are you setting up a whole separate vhost instead of just adding the second domain name as a ServerAlias on the existing vhost?