The town of Stockton is all in Durham. The council area of Stockton on Tees does cross ceremonial countiesThe Stockton on Tees, the only town in two ceremonial counties, Yarm although a surburb was a port once upon a time, so its Yarm /Ingelby/ Thornaby on Tees imho.
The only bit that was South Stockton is the area between the old river and the new cutting of the Tees. Basically the golf course and racecourse (Teesside Park shopping)Thornaby was and still has its village, the rest of it was Stockton South Historically.