Here's a way to do it.
I remembered reading about a way to make Anonymous modules disappear for registered users, but it involves a hack — changing a couple of lines in tikilib.php and userlib.php. Did a search at the SF forums and found it. See this page
about separating Anon and Registered users (scroll down below the warnings 😉. I implemented this at a site and it works fine. Now my registered users are no longer Anonymous and so modules, etc. targetting the general public don't appear on pages for people logged in.
-- Gary