Stoke-on-Trent is comprised of six towns. North to south it is Tunstall, Burslem, Hanley, Stoke-upon-Trent, Fenton (my area) and Longton. It is Arnold Bennett who is to blame for the confusion, his novel Anna of the Six Towns was renamed by the publisher who thought Anna of the Five Towns sounded better, so he left out Fenton.
Hanley is called the city centre by the council despite a lot of opposition from the public who still consider Stoke itself to be the city centre. Hanley is the commercial centre. Stoke was at one time the only city in the UK which did not have a cathedral. King George V conferred the honour in 1925.
The western boundary for the City of Stoke-on-Trent is more or less along the A34, from the junction with the A500 to just south of the junction with the A5035 at Trentham Gardens. Trentham Gardens does have a Stoke postcode, but so does Alsager which is in Cheshire.
Just as confusing is the Wedgwood factory at Barlaston. It is actually in an area administered by Staffordshire Council along with Trentham Gardens. So any tourists who say they have been to see pottery made at Wedgwood in Stoke, technically have not done so.
Yes, I was good at history and geography in my school days.