This describes Pitt as "Private, with some state support."
Other sites scattered about the internet call it a "public research university."
I should admit, though, that I was described Temple and Penn State, and extending that description, based on anecdote, to Pitt. And the difference between state and public is this: state schools are funded primarily through state funds. tuition is very very low. Like 6K/year and less at most of the schools (see the pennsylvania state system of higher education, I think it's called). the state-funded national public universities, as you noted, get some state funding, but are large research universities. that they get some of the state-funding is one of the reasons tuitions are kept relatively low (compare pitt's cost with CMU's, a truly private institution), and it is also the reason the schools are considered public institutions. so, if it is true that pitt is in name a private school, it is de facto a public school. or so most folks will characterize it.