I've have zilch experience of private schools, so I can't answer dinosaur's question. I wish our state secondary schools were like the one suedonim mentions, or we had a choice (even if was flawed) of some good local state secondaries like janh has in her area.
I have looked at your list, ks, in your most recent post. I could write a very similar list for my son's church school.
We don't have a house system though there is a system of bonus points, but perhaps not so much fostering team spirit.
No, children aren't told to shake hands with people on first meeting, but they are expected to hold open doors, say 'hello can I help you' to strangers in the school etc.
Effort is rewarded very publically at the school, too. Each friday assembly, all children who have gained any award outside school - sport, music etc - can go up to collect it from the headmaster. These aren't just for activities directly connected with the school, parents let the head know and give him the certificates to present. Sometimes it seems half the school are up collecting something or other.
I can contrast this with the two state primaries my son attended.
I would say there was less emphasis placed on publically and formally recognising effort - no long assemblies full of award-giving.
But one thing make is really difficult to form comparisoms and that's size. The state primaris were 2 or 3 times as big as my sons church school. The church school has 22 -24 children per class as opposed to 32 - 34 in the state school.
Consequently, I feel there is a closer community at my son's much smaller church school. Áll teachers know all pupils by name at least. All pupils know each other very well. Most parents who are active at the school know most of the pupils - not just those in their child's class. There is a strict uniform code. All pupils, from reception to year 6, share some of their playtime together and share the same playground. Most school activites - sports day, music events, religious events, etc are shared by the whole school at the same time. The only separation is between infants and juniors.
As a result, things like social confirmity, sense of team spirit, manners, sense of responsiblity towards others inside the school, confidence when talking to adults within the school, are IMO more in evidence than what I saw at my son's state schools. This difference isn't always for the better I hasten to add. But lots of the difference boils down to size - and the leadship from the head. And the head has more power at this small school, even though
their 'empire' is smaller, precisely because he knows everyone so well and his influence can reach further.
Dinosaur, if I was to pay for private education it would be size of class, and ratio of adults to pupils that would be the main attraction. In my limited experience, this factor has far reaching effects.