I live in Whitchurch, Hampshire, which is technically a town (the smallest in Hampshire) and has a population of about 7,200. "Local" to me means Whitchurch itself and the surrounding countryside for 3 -4 miles. Four miles eastwards and you've reached Overton, a large village of a bit under 5,000, which generally I don't consider "local". We have four small food stores, one of which contains the Post Office. None of them is large enough to be called a supermarket. There's a baker, a chemist and a petrol station. But there isn't a butcher, and if I was asked where there's a local butcher I would say there isn't one, but the nearest one is in Overton.
"Local walks" would be ones that I feel I can do on foot there and back, or maybe up to four miles if I can get home by bus. That runs Andover - Whitchurch - Overton - Basingstoke. It's "local" for about three miles either side of Whitchurch. Plus the remains of a previously more regular service southwards to Winchester, largely used by students. There's a genuinely "local" minibus service on three days a week operated within Whitchurch, which also runs once a week to Morrisons in Basingstoke, about eleven miles away, and allows people to spend a couple of hours there.
Administratively, we're in the District of Basingstoke and Deane, but I don't a consider Basingstoke to be a "local" large town. Andover, in Test Valley District, is only six miles away, but I'd call that the "nearest" large town rather than a local one.
Congratulations to anyone who can deduce any coherent principles from all this!