When you are trying to define an institution which imprisons people for riding a train without a ticket, then I think it’s important to dissect the word “dangerous.” I would not consider them dangerous.

I agree that person is not dangerous. As I've pointed out, the fact that non-dangerous people are in prison does not keep "dangerous people" from being a defining characteristic of prisons.

All prisons must have X, so something that includes X + Y can be a prison even though it doesn't exclusively include X.

schools have entire police forces, and have apprehended kids who’ve skipped and brought them back

That's closer to a prison, but as the school isn't trying to separate dangerous people from society -- you can get expelled for starting fights, and school isn't anywhere near as separated from society as prisons -- it's still quite a bit different.

It is a propagandized definition.

My definition is the bare bones of what prisons do in practice. It's the core concept, it's the essential elements. There are prisons that do not wage class warfare (see the Soviet penal system, for example), therefore that is not an essential element of the definition.