Hi, Megan, this sentence "Wasser bitte" is so kryptic and close to inpoliteness that it is not to recommend to use it. It would be better to say: " Kann ich bitte Wasser haben?", but that is more than the lesson allows. In this case, Duo really wants Wasser bitte = Water, please, Just try to see it as an exercise = Mach' eine Übung draus!
Yes, it should be, but I think there is a nuance. Only water please suggests Only water please, and nothing but water. It seems just a bit more directive. Just water please suggests a degree of unconcern or acceptance Just water please, I don't want to make a fuss, I'm quite happy with that.