Refer to the WTC2 and WTC5 documents, not Gov.uk. The info on Gov.uk is too simplistic, HMRC has 2 booklets about this so Gov.uk only give a brief summary.
What I think they are trying to say is that you can't use tax credits towards school care or care provided at the school which is run by the school and is not registered separately to the school.
You can have a nursery on a school site, in partnership with the school but it would have its own registration. A reception class in school would not have a different registration to the school. So a 4 year old in the nursery can be funded using tax credits, whereas a 4 year old in reception class can not.
In an independent school, childcare vouchers can be used for children in reception class. Not sure about tax credits though, doubt it every comes up as private school fees may be 3000 a term.
A summer cub run by the school, at the school, may not have a separate registration.
I think the key thing is does whatever you are considering using have their own Ofsted registration?