BigPinkLad. Not sure how you would separate the two geographically, in the sense that there was no moat or diverted river dividing Newcastle from Northumberland. However, if it had its own political rule that owed no allegiance of any kind to Northumberland, then it wasn't part of Northumberland. These names are only human constructs, not written into the physical landscape.

Similarly Dr. Quelch. You might think it somehow "right" that the Irish should be British, and for the time they were ruled from London you would have a point. But they are not now. The term British is politically neutral and does not include the Irish. The majority in Northern Ireland might prefer that it did, but it doesn't. The term Brit carries negative political overtones and definitely does not include the Irish! The language is as independent of external diktats as the sunrise.

I'm not sure how the history of the County Palatinate of Durham, which did not have the same borders as the current county, links with the distinctive use of "County Durham", if the term is not used to clearly distinguish the county from the city.