I also thought that a real 'waiter's friend' was the two prong type without a corksrew. So named because it didn't damage the cork upon removal which meant the cork could be put back.
I've heard that one referred to as a 'Butler's Friend' Uncle, for the reasons you site and additionally that the bottle could topped up
The one I bought was this:
Looks a bit manky in the photo but works a treat.
This one:
is likely to cause damage; tight as a gnat's chuff!