the way I see it, and I may be wrong, is that there are different levels of thirst, so with Water Fasting some people may drink sooner than others depending on 'how' thirsty they feel.
With dry fasting you just go for as long as possible with no water and carry on a bit once it does arise.
I find that when I do feel real thirst(dry mouth) while dry fasting and I then try to turn the fast into a Water Fast I can't , my body doesn't like it and considers the fast over.
So with Water Fasting I think you must have to drink when the first level of persistent thirst arises, and with dry fasting you only drink once your mouth goes dry which is extreme thirst.