If I ask for a cup of ice water, I'm almost always given a small cup of water with a couple of ice cubes in it. I will occasionally ask for a large cup of ice water.
We always bring bottles for water. My most recent trip, I didn't bring one. I wanted to try just stopping whenever I could and asking for a cup of ice water. I didn't last a day, and ended up spending a ridiculous amount on a cute bottle at Epcot.
You don't need to buy anything to get the free cups of ice water. I will typically get in a QS line and just order a cup or 2 of ice water, unless I notice a tray of cups already filled, then I will go directly to the counter and ask there.
The don't have it at carts - just QS locations. And it's good, filtered water. I like the self service locations best (BOG at MK, Sunshine Seasons at Epcot, Backlot Express at HS, and Satuli Canteen at AK - there might be others, but these are the ones I used on my most recent trip). I would also fill up at the resort whenever I was near the food court.