Can't speak for Carlisle but i'd regard York as North East. In the grand scheme of things.
You wouldn't find many of the citizens of York would agree with you - they are Yorkshiremen first & foremost., although not in the same way as, say, Leeds people are(people from W.Yorks brag about it more). To me, the North East probably ends at Scotch Corner in the south, Bowes in the west, and Berwick in the north.
Aye, North East is County Durham, Tyne & Wear and Northumberland.