A library is everything in the iTunes folder. The best solution for you is to ensure the entire library is on the external drive:
Apple's 6 pages of exhaustively meticulous step-by-step instructions: How to move your iTunes library to a new computer - https://support.apple.com/HT204318 Use this if you suspect any of your media are not in the standard location (iTunes Media folder).
My one-sentence summary if you always use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its sub-folders and files) intact to the other drive, hold down the option/alt key (shift on Windows) while starting iTunes, and when requested to select a library select the iTunes Library.itl file in the copied folder (Open a different iTunes Library file or create a new one - https://support.apple.com/HT201596).