"Family History and Genealogy sites for classes and research only." It sounds like you're trying to exclude something. What about inputting names into newFamilySearch to generate temple ordnances? Or is that considered research? You might want to check the language in the CHI to see to what extent admin computers are to be made available to members for genealogy work to make sure your guidelines are in harmony with the handbook.
In looking over the Internet Use Policy that was dated 2006, I found something you may want to include:
No Privacy Expectation for Internet Access: Internet browsing though Church systems is filtered, monitored, logged and reported on a regular basis. The ability to connect to a specific web site does not in itself imply that visiting that site is appropriate. Users should have no expectation of privacy when accessing the Internet though any application.
I am no lawyer, but "no expectation of privacy" sounds significant.
Likewise from the FHC Internet Use policy sign:
Users should have no expectation of privacy when accessing the Internet using family history center computers.