Someone recently found my blog using the search terms “Capital L library?”. It might be too late for them, but here’s my interpretation for anyone else seeking an answer to this question.
Library-with-a-capital-L is a Proper Noun and refers to a named library such as St Hugh’s College Library or the Bodleian Library, which may be shortened to “Library” when referred to multiple times in the same document. When referring to a library in a general way, little-l-library is appropriate.
What about Librarian-with-a-capital-L? For most people, a librarian-with-a-little-l is a person who has a front-of-house role in a library and helps you borrow and return your books. However, front-of-house roles are normally carried out by library assistants, while a Librarian-with-a-capital-L is an information professional with a postgraduate qualification in the field, who is more likely to be found doing staff management or budget planning than checking in a book.
Among Oxford L/librarians, this distinction is well understood, so I can introduce myself as the College Librarian and people know that this means. Away from Oxford, I tend to say that I am the Library Manager, or something similar (Empress of Information? I should try that sometime).
I think the subtlety of the L/librarian distinction also explains why some people find it hard to believe that I need any qualification at all to be a Librarian-with-a-capital-L, as they think I said librarian-with-a-little-l, and why would you need an MA to shelve books?