I do this all the time. I'll always ask for "A large cup of ice water, please." No tiny cups here. They never say I can't have that, or even so much as look at me funny. In fact, when I was MK last month, at Storybook Treats I ordered a large cup of ice water, and a cup of vanilla ice cream... and the CM didn't even charge me for the ice cream!!! (Yay, pixie dust!)
I think that WDW would rather their guests be well-hydrated and not have them passing out all over the park, than quibble over some free water, honestly. Don't be shy, get your free water on. (And it's always tasted just fine to me, too!)