Most, if not all, pirated books are copies of books that had been legally obtained, so this is not how they are distinguished from books borrowed from a library. The only thing that makes them pirated is that the price paid for the original book is considered to not have covered the right of also distributing copies of the book.

Nowadays the surviving public libraries might pay special prices for the right of lending books, but that was not true in the past, when they just bought the books from the market like anyone else, at the same price.

I am pretty sure that the public libraries that I frequented as a child, many decades ago, did not pay anything for a book above the price that I would have paid myself, but nonetheless at that time nobody would have thought that they do not have the right to lend the books to whomever they pleased.