Some of us keep food and water inside the coop only. Some of us keep it outside only. Some of us keep food and/or water both inside and outside. There is no one right answer for this that suits all of us. We all have different situations and our own reasons for doing what we do. What works for one may not work for another. I'm only going to scratch the surface of all the different reasons people do what they do.
Some typical reasons for feeding in the coop include people don't want to feed wild birds if wild birds can get to it outside. If you like to sleep in on a weekend so you are late letting them out they can get to food or water. Many of us lock them in the coop section only at night for additional predator protection. If you consider your run predator proof you don't have to do this. It may be easier to keep water thawed in winter and feed dry inside.
Some reasons to not feed or water in the coop is that water can spill and get the coop wet. Feeding inside the coop can attract rodents like mice and rats inside. The more time the chickens spend in the coop the more they poop inside. You may have to manage the poop more if they are inside more eating and drinking. Some coops are so small it is hard to set up roosts so the chickens are not pooping in the feed and water from the roosts. Feeding and watering in the run changes some of this.
I feed and water both inside and outside. I am sometimes slow to go down there and let then out on the morning so they have food and water. But the main reason is that I am always integrating juveniles. Having feed and water available in multiple places makes integration a lot easier, the older ones can't bully the younger away from all the food and water. Since mine have an open grassy area behind electric netting to graze on I can't keep wild birds out. That gate has to remain open. I sometimes find wild birds inside my coop.
Choose what makes sense for you and go with it. There is no really wrong answer.
