"Also, I want a cup of water."
Translation:Επίσης, θέλω ένα ποτήρι νερό.
Ποτήρι του νερού (with definite article)=water glass (glass made to drink water from it)
In Greek, just like for example Italian, the verb ending always indicates the person, so you can always omit the subject, unless you want to specify it. For example, if you want to stress that you, not someone else, wants the water, or to distinguish between αυτός and αυτή in case you are with multiple friends.