And I'm sure nobody is swigging the three ounces from that little cone and then refilling their used cone......the water bottles (like mine, where the part I drink from comes off to fill it) aren't the only possible contamination. However, they would drain that dispenser much more quickly, causing more work to refill it. I'd say it's a coin toss as to which one is driving that sign.
Personally, I take my own bottle and keep it filled with ice and water all day. Even if they put in filling stations like in Pandora all over the parks (which I think is a great idea), they will likely still be warm. I'll just stick to getting counter service ice midway through my day. I'm not a "no bag" person anyway, and I always carry a bottle, not just in the parks, so I'm used to if.