Posh to me is something a bit more, a bit nicer than the norm, a posh frock rather than the usual jeans and TShirt.
Driving through an affluent area ‘wow there are some posh houses here’.
A restaurant where the food and service are extra special, silver cutlery bone China crockery and crystal glassware might be a bit posh.
Usually more expensive might be considered posh but not always. To my mind posh people are those with a little breeding who go out of their way to make others feel comfortable and welcome, often live in draughty old houses with dog blankets over the stuffed armchairs and drive rusty old bangers.
Schools are either good or poor, whether in the state or public sector, so I can’t see how a school can be described as posh.
Just my humble opinion.