Oxford = private institution?

In the DFes-Form part 8 it says to tick the box if you are planning to attend a "private institution".. Well, I have no clue - is Oxford "private" like most American universities or not? In the guidance notes it says that you can't apply for financial aid in this case:

"If you are attending a course at a private university or college which is accepted by this Department, you are not subject to a financial assessment and are entitled to receive the full tuition fee contribution. You should tick the relevant box then go to the declaration at Part 15."

(from http://www.dfes.gov.uk/studentsupport/uploads/EU5N-GUIDANCE-NOTES-2005.06.pdf)

So I assume it's not private in this respect coz you could generally apply for assistance, right? So is Oxford really run by the government.. ? :confused: