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More than a book collection, less than a public library, the personal library has a number of traits that sets it apart.

A personal library is owned by a person. I don't think families can own a personal library -- it must be the primary haven for one person's mind. A family can be home to more than one personal library, but it would be rare.

A personal library is somewhat like a public library, in that it might be organized by the Dewey Decimal System. It might have a catalog, either a card catalog or something automated. And it might make loans, but not to just anyone. Friends, generally, and only friends who know how to treat a book properly.