The CoA used for Durham in CKII is that of the city alone, and I'm not sure how old it is. Quite a few flags for local authorities were only just invented in the 70s, like Cumberland's.
This CoA has definitely been around for centuries though, being the CoA of the Prince-Bishops - I've seen it myself on crests on the old buildings in the cathedral square - and a slight variant is used by the county council even today: https://upload.wikimedia.org/wikipedia/commons/1/1f/Angl-Ebor-Durham-Arms.svg
Plus, the CoA I linked is just a better fit for the county as a whole, rather than just the city.
This CoA has definitely been around for centuries though, being the CoA of the Prince-Bishops - I've seen it myself on crests on the old buildings in the cathedral square - and a slight variant is used by the county council even today: https://upload.wikimedia.org/wikipedia/commons/1/1f/Angl-Ebor-Durham-Arms.svg
Plus, the CoA I linked is just a better fit for the county as a whole, rather than just the city.