Well, I'm not sure if I'm able to shed any more light here or not. I've created a new test page with nothing else on it but the three rules in question, and had the client look at those. Essentially, the "show region" works with the standard "logged into users" "not logged into users" rules, but not my rules created to check if he is level 1, or level 2+. However, on my system, using his login credentials, it works perfectly fine.
I've attached the new test page with comments as to which works and which doesn't. I'm also providing his login credentials in the private message.
Is there something with the <= 2 rule that would cause it not to read properly on some devices (being server side, I wouldn't think so)?
Where are these rules defined in the site? Is it possible that that rule was cached at some point and then overwritten, so that anyone who visited prior to the overwrite would have the file cached that defined the rule, but a later user wouldn't (I use two different systems, so this is a possibility but I'm not sure where those rules are defined to verify)?