Right, so a "public" library is a good example of a good that is provided publicly, but has little economic justification as such. A "public" good is technically specific in economics, and refers to something more narrow than what is used in everyday language.

A book is excludable, even if somewhat nonrivalrous. It's rivalrous in the sense that it can't be checked out to multiple people at once, but nonrivalrous in the sense that a book in a library can be consumed by many more people than a book kept on a shelf in someone's private home, over an extended period of time.

A library could operate without positive external effects with a subscription model.