Dear,
This post might be a little outdated but I would still like to contribute to the discussion.
Apple Music does have a source, and it's a user-contributed community dubbed Musixmatch. It's like Wikipedia for lyrics, seriously. There are levels though (Beginner, Curator, Specialist) you have to reach for some songs, because it would be quite chaotic if everyone could mess with lyrics on all songs. Despite the "user-contributed" being its attribute, it functions surprisingly great.
They also provide lyrics to Spotify, Shazam, Vevo, Instagram and others. Leveling up over there is quite easy! I'm a curator myself, just under a month of some song listening and syncing you can too, but I doubt it you would really have time or energy to do that if you only want to change a few words in a specific song. (ask me if you want me to check up on a song 😁)
Anyways, if there are lyrics that are wrong, at any time you can jump over to musixmatch.com or install their app to check whether it's unlocked or someone with higher rank has edited it previously. In some, but very rare cases (probably 5%-10% of all songs), edits on MM will not show up in the Music app, because artists/record label companies have added their own. In that case, just follow the previous reply!