I like having water on me all the times if I can help it. You mention that drinking fountains don't bother you, do you mean specifically the fountains at WDW or are you speaking generally? Most of the fountains at WDW are straight taps, as opposed to the style of drinking fountain that goes through a refrigerated chiller that are more common out in the real world. The water from WDW drinking fountains always strikes me as warm and ... off tasting. The water they give you from a soda fountain is ice cold and has passed through at least a simple filter.
So, cup or flat bottle? ... What advantage does the cup give you over simply taking a drink from the fountain? You can take some water with you as long as you finish it before the next ride or whatever. Okay, can do that with a bottle then also seal bottle up tight and carry through a ride or whatnot. Cup is easier to fill in general, much easier if filling from a QS cup. But not really easier than the bottle if filling from a drinking fountain. If water pressure is low, the bottle may fill easier as it can lay on its side. If you plan on filling from QS cups, a small neck bottle would be a real pain.
There are wide-mouth collapsible water bottles. Some are the flat, envelope design, some are like regular short water bottles that accordion out to full size. Of the latter, the "que Bottle" or one by Skyyhi look interesting.