Did the York class improve on County-class protection?

Did the York class improve on the earlier County class with a proper armour belt?

No. The Yorks were Type B heavy cruisers intended to save treaty tonnage, not a materially better-protected design.

The premise that the York class introduced a materially better protection scheme than the earlier Counties is rejected. Both classes were pre-war designs, so the Yorks cannot be treated as a wartime correction to an obsolete predecessor. They were the Type B economy design, intended to save treaty tonnage rather than spend more displacement on protection. Their reduced form was a cost-and-tonnage compromise rather than an armour upgrade.