You are quite right, Truck. Durham city stands astride the River Wear. The Tees, further south , is a Durham river only
by an extension of credulity or symantics.
Remember a song by Roger Whittaker, 'I'm Gonna Leave
Ole Durham Town' ? The lyrics ran:
'When I was a lad, I spent my time
Sitting on the banks of the River Tyne.'
It strikes me that he would have had a long way to walk, to sit on the banks of the River Tyne.