I think it’s a County Council. It is a Council that covers the area of a County, and it’s called County Council. I guess what’s fairer to say is “Labour holds none of the county councils that are on the main county electoral cycle and which voted this week”.
Also in law, Durham is a County Council – all unitaries are legally either ‘district councils performing the functions of county councils’ or vice versa. Admittedly the law isn’t the strongest point, since Bristol is designated a county and Herefordshire designated a district, but never mind.
I don’t think so – a County Borough was a County Town with unitary status outside the County of which it was the town. They were abolished in law in England and Wales in 1972, though obviously some places are de facto County Boroughs, like Leicester or Derby. The new Durham County Council is for the whole County, not just the City.