This is one of those things that comes down to what works best for you, your flock and your setup - it's really just a personal preference for which there is no one right or wrong answer.
What type of feeders are you planning to use? This will factor into whether it is something you would want exposed to weather conditions (ie out in the run) or not - treadle feeders. closed "no waste feeders" etc would not be an issue for food spoilage from rain, etc as much as the more open feeders would be. If kept outside, the more open feeders would require some additional thought and construction to provide cover over them (many folks hang them outside under a raised coop so that the coop floor provides the roof over the feed/water station).
How enclosed is your run? Are there other animals that you would need to consider who may access the feed/water station if it is outdoors and what, if any, possible risks can you see that causing? Example - the horses in your avatar - poultry feed is grain based and quite attractive to horses, but not something the ought to be consuming. If you leave the run open to allow the birds to free range it would also invite the horses to do a little exploring (speaking from experience as I have one very snoopy horse who has tried, on more than one occasion, to wander on into the run while the door was open).
Water in the coop can lead to mess if it is bumped/spilled and moisture in the coop from that spill can contribute to overall unsanitary conditions, set you up for moisture loving disease conditions, contribute to the humidity of the coop (important when considering frostbite risk during cold winter temps, especially in a poorly ventilated coop) etc.
What is your overall climate and how likely is it for there to be days/weeks when the birds would not want to be out in the run?
How accessible is the coop going to be for you to be able to get to and fill a water feed station?
How accessible is the run going to be for the same?
Are you already pushing the limit for space per bird inside the coop? (depending on the type of feed/water containers used you can take up a lot of "floor space" having them occupying coop space)
Will your birds be locked into the coop at night and, if so, when will someone go out to let them out in the mornings?
There are a lot of things that can be considered and arguments to be made for either or both - I'd say, pick one that sounds right for you, try it and, if you don't like it, try the other until you find what you like best.