"The dog drinks his water."
I didn't get your question, since I don't see all the other sentences and I am not taking that often the tests in Italian (even if I tried... to see how it was).
Anyway, I would translate "his water" into "la sua acqua" and "his own water" into "la propria acqua".
Here, since the dog usually should have "it" as pronoun, could even be stealing the water to somebody. :D
I link you here something regarding "proprio" to translate "his/her/its own"...
My problem is the following: Sometimes the translation of "proprio" is translated to "his own", sometimes it is translated to "his". But is not consistent with the use of the term. To get back to my example, I had the sentence "The dog drinks his water" -> "il cane beve la sua acqua". But at another point in time with a similar sentence "The boy eats his [...]" -> il ragazzo mangia il suo [...]/il proprio [...]". I don't mind being wrong since that's a way to learn but I would like some consistency. Maybe I can find it with your link, thank you.
The direct translation of "his thing" is "la sua cosa", "his own thing" is "la propria cosa"...
But, you know, translating is something borderline, and there are no precise translation, and the most of duo is updated because of suggestions of the users, therefore some changes are reflected on some sentences but not in some others.