I know that facebook allow me to specify multiple domains for my app.

Bu
As long as you listed all the desired App Domains in application settings you should be able to authenticate users on any of them.
"Website with Facebook Login" is really only intended to be used as link to your site/application.
Update 2 (July 2016):
App domains must match the domain of the Secure Canvas URL, Mobile Site URL, Unity Binary URL, Site URL or Secure Page Tab URL.
Update (December 2013):
At the time of writing original answer it was possible to list any domains in application settings but from that time the UI of Application Settings (as well as way of handling Application Domains) changed at least couple of times, at some point you could only list domains that derive from one of application's canvas pages.
As of December 2013 it is possible (again) to list domains that do not derive from application canvas URL.
