...Pietro Frua was head of design at Ghia, having sold his facilities to fellow Turinese carrozzeria, Ghia.
It is important to distinguish between Frua the man and Frua the coach building company. Pietro had did not design the P1800, but the facilities that he once owned built the prototypes.
As for naming the design, the 1958 Renault Floride is credited to Pietro, but is considered a Ghia (design house) design. And if we apply the naming convention for say, the VW Karmann Ghia, we would have a Volvo Jensen Ghia or is it Frua?? Confused, yet?