in NA we have to call them "free libraries" because the concept of not having to pay for something is so rare, it has to be clearly indicated upfront! (joking lol, mostly!!)

I think being free is implied in being a library

Some libraries require membership. Also some people have private libraries. Universities often have libraries which, at least, don’t offer all of their services to the general public.

Not being open to the general public doesn't mean it's not free

I’d argue it is paid for by the students’ tuition, but money is fungible so I’m not sure how exactly we could nail that down.