TerfTerfTerf I don't know about college owned rooms in Durham, only about private lets. I'm not sure that colleges other than those on the Bailey do own rooms outside the main college building, so college arranged seems more likely.
The great thing about Durham is that it's extremely condensed - I'm struggling to think of rooms which would be outside Durham itself, I don't think that will happen. So the chances must be that your DS would be within the City, unlike some horror stories which came in about eg Bristol last year, placing students across the Severn Bridge.
I also have to say that in my experience (probably too ancient to be of much relevance) as well as DS4's (who left this summer), college choice or allocation is supremely unimportant at Durham. Your DS is doing a fun subject and isn't tied to labs etc so plenty of scope to try out new things, assuming the uni doesn't go over the top about restricting activities in the name of keeping young people safe.
I'm very biased (aka enthusiastic) about Durham. I don't think there's any contest between York and Durham, either as a place to be at uni or in terms of PPE (but I'm fairly superficial - I'd go for place over course any day :)).
Good luck to your DS.