You had almost completed the process successfully. Had you simply followed through, it would be done. Not to fear, it's easy to continue and finish it.
You are correct to 'Edit the library' means exactly that. Hover over the edit pencil for the library and click it; go to the 'Folders' tab; Add the new folders/directories.
You will see the number '2' in the upper right corner. This shows that Plex now sees the same movie in two places; the old (which hasn't been cleared out yet) and the new (which it just found and cross referenced into the existing metadata)
The error you saw when attempting to play the movie after renaming the folder is because you got slightly out of sync. Plex was still looking for the 1st copy before looking for the 2nd copy.
After adding the new location, which puts the '2' (instance count) in the corner of each poster, when refresh/update is complete, go back to the same library and remove the old folder (so it is no longer referenced) then refresh/update the library one last time.
What this does is remove the '2' (the reference to the old location) from the database.
After this is complete, you can empty trash/clean bundles as well. Now all references to the old media location are gone and everything will work as before.
Here is another way of doing the same thing.
1. disable / turn off "empty trash after library update" (if this is enabled) in Settings > Server > Library
2. make the new media location available to the computer
3. edit (add) the new folder location to the library
4. copy the media to the new location
5. refresh/update the library (which will cause the '2' to appear)
6. edit (remove) the first folder location from the library
7. refresh/update one last time
8. verify all media correctly accounted for.
9. empty trash / clean bundles
10. delete the original copy of the media (the old location)