"Fake" seems fine. If I buy a fake watch, that might mean that it's a real watch that does its job of telling time, but it says "Rolex" on the front and that's a lie.
"Fake" seems fine. If I buy a fake watch, that might mean that it's a real watch that does its job of telling time, but it says "Rolex" on the front and that's a lie.
In that example it would be more common to describe the watch as a "fake Rolex", for the reason you give (it's a real watch).
The proper term in your example is “counterfeit”.