You need to have the same realm set for both domains, such as:
$openid = new LightOpenID('www.example.com');
If you use different realms (domains), it is valid for a provider to return different identities, as Google does, for example.
Also, you can't rely on the email address returned by OpenID without further verification. From the protocol point of view, it's just a string value that's filled with whatever the user wants.
And lastly - don't limit your OpenID support to Google. The protocol was specifically designed to make it easy to be decentralized, and it isn't much harder to at least provide a way to enter any openid url, instead of just clicking "sign in with google".