Based on the government approved types of school, what type of schools is Hogwarts?
I believe it would be somewhat like a Foundation School where it is state funded and has a governing body that technically owns the grounds.
None of the above. The school is evidently funded by a separate non-governmental body that exists within the UK but isn't answerable to the UK government. On that basis (and taking into account that the school is neither licensed, nor inspected by any Local Education Authority nor OFSTED) I would have to say that the closest description of it is that it's not a school in any discernible sense of the word but rather a private club for children of school age.
Because education is definitely taking place and because none of the teachers hold QTS status, you could possibly argue that it's a private academy of some sort (if you squint and turn your head sideways). It would certainly be illegal to operate on that basis so at the very best it's an illegal private academy.