If its a STATE institution, it’s public. If it is not state-run, it is private.
All of the best institutions are private. They are also a lot more expensive, since they don’t get support from public tax money.
BUT… for a foreigner, there might be little difference in cost since the non-resident tuition and fees for a state-run school are much higher than for residents (who are taxpayers).
State-run schools tend to be much bigger, since their mission is to provide a college education for everyone in the state who wants one. That means they have very large classes and rely very heavily on online instruction. It’s lots cheaper to have one instructor for 300 students instead of one instructor for 30 students.
I went to a good private college. Most classes after freshman year had less than 20 students in each class. I see similar courses taught at a state school where classes are so packed with students that half don’t show up and they STILL have 300 in the classroom. In such courses, the “teacher” does little more than administer an online course.