As the text says, the wizard has the cloak, but is not an "invisibility" one. Instead; is just a cloak that can't bee seen by the human eye . It was a silly play on words that serves as a minor hook that the players may be interested in. The cloak is intended to be the one that was stolen from Acuta.
I agree that maybe it is a little confusing, so I will clarify this in future corrections. Thank you so much for your input.