>> "My question is: Does this contact with the LimeLM servers "convert" the license into an "online" activation thereafter?"
Yes, only if IsGenuineEx() is scheduled to talk to the servers.
If the customers have a consistent Internet policy that is actually secure (i.e. never allowing a computer that's not supposed to talk to the outside Internet to talk to the outside internet) then this won't be a problem. It's the fact that their policy is broken that's causing problems.
So, solution #1: the customer should fix their broken Internet policy. It's not secure.
Solution #2: You develop a "special version" of you app that uses IsActivated() rather than IsGenuineEx() for customers that absolutely cannot have internet-verification of your app (even despite a broken security policy). IsActivated() always checks the license locally, and never re-verifies with the activation servers.
Honestly, solution #1 is the real answer. Solution #2 is really only a solution if the customer is especially stubborn.