I know York very well. Grew up there, still a regular vsitor and toured all the private schools a couple of years ago.
St Peter's is a very good school. Judged by many to be the best in York. Good league table position if that is what you want but also an old fashioned boarding school tradition and successfully making the rapid transition to being a day school with a much lower number of boarders.
I do not know about the level of strictness but discipline in a good thing in my view. We are seriously considering sending our DS1 and DS2 there.
They appointed a new headmaster recently and my only slight query is that I get the feeling that the school is perhaps losing its boarding school ethos which is one of the thngs that attracts us to it - even though our boys will be day pupls. Parents are typically well off locals plus Notherners who have moved to London and sent DCs back to York for education plus a smattering of UK expats, forces parents and foreign parents who want a traditional UK boarding school education.
Your alternative in the private school sector is Bootham whch seems a bit more relaxed but lower league table position, less sporty and less of a boarding school feel and the facilities are less good. The parents are much more local, a lot of University staff send their DCs there as they have what amounts to a means tested fee policy.
You could also think about Pocklington school but do not know much about it. Day schools not really known to me.
We tried to move to York a few years back but failed to get there as we simply could not find a house at a sensible price. That is going to be your biggest problem. It is a lovely place which is why a lot of retirees who are Northerners but who made their money in the South East / London come back and push up prices.
There is it seems a bit of an economic slump going on in York with a lot of empty shops and offices that I noticed when I was there a few weeks ago. House prices have moderated a bit but it is only a small decline. It will always be a place people aspire to live.