Excusing the beers for a moment, that's about the dumbest thing I've ever heard regarding our W113 series cars. Yes, the "Pagoda" nickname derived from the shape of the hard top. Of that there is no question. The nickname refers to the W113, NOT whether or not it came with a hard top, or whether it came with a soft top (California Coupe), or whether the soft top is up or down, or if the hard top is on or off.
Whoever you were drinking with needs an education.