Library / Collection = different terms, same thing... the "collection" is your music, it's all stored in the "library." The database is just the file in which all of this information is stored.
To move a database, you can do so using the move technique outlined here, but if you want to make a copy, you're better off to use the File > Library > Backup Library on the source computer, then Library > Restore Library on the destination. And for this you're pointing to the ZIP file containing all of the contents, not just the database EDB file. That's likely why you didn't wind up with all your data being found.
Let's review... you say you "backed up" and then "moved" it ... those are two different things. Give me the step-by-step of exactly what you did.