Sorry, but this is only partially correct.

liber as an adjective means free, this is correct. But the origin of library is the noun liber (book) which is used metonymically only. People used to write on bark, which is the correct translation of the noun liber. While this meaning of bark is related to trees and not to dogs, the meaning of library is related to bark and not to liberty...

Damn, langugages are difficult and confusing.