This same question has arisen in other similar exercises, but just to confirm "Το ποτήρι του νερού" can mean both "the water glass" (i.e. "glass that is used for serving water") AND "the glass of water" (i.e. "a serving of water contained in a glass")? This is like how "το πιάτο της σαλάτας" can mean both "the salad plate" and "the plate of salad," right? I guess I would like to know if there is another way to distinguish the two meanings...for example, you are in your bedroom and you call out to your σύζυγος (who is in the kitchen) to bring you "ένα ποτήρι του νερού"...would he/she likely bring you an (empty) water glass or a glass of water? I am assuming (perhaps incorrectly?) that the latter is the default interpretation... but how would you specify the former? (i.e. you just want a water glass with nothing in it)
For me (a Brit) "The glass of water" is a glass containing water (a glassful of water). Surely that would be το ποτήρι νερό ?
All the entertaining explanations above seem to be saying that το ποτήρι του νερού is a water-glass (what the French call "un verre à eau" rather than a "verre d'eau") and would, by default, be empty.