Ipaq, you are right.
CAT III dual means, that your systems are fail/operational, that is in case of a failure, you can continue after getting the click,click,click, and CAT III single means, you get the warning and have to abort. The system is fail/safe. The reasons for it are legion, Elec, Hydr, steerability, autopilot, IRS, just check the table...
Nic