Not all non-alcoholic beverages are included in the dining plan. It includes the basic fountain drinks such as coffee, tea, iced tea, soda, lemonade, etc. - essentially beverages that fall into the "free refills" category.
Some Disney owned restaurants will allow milkshakes, smoothies, virgin cocktails like the Shirley Temple, pressed pot coffee. We've gotten all of these as the adult's dining plan beverage. You'll have to ask at each restaurant if you see something on the menu. The milkshakes and smoothies cannot be ordered on the children's dining plan; I don't know about the Shirley Temple.
The third party restaurants offer only the fountain beverages because they won't be reimbursed by Disney for anything else. If you want the Agua Fresca at VN you'll have to pay OOP.