Not sure about your instructions on reproducing the issue as the Album tab doesn't show Artist's names, until you click the Genre button, and then it shows the artists names correctly. The spot where I saw an error was in going to the Song tab.
From a brief look at what's happening I believe this problem is because you are displaying the artist name by the NSString saved into your song called songArtist as opposed to getting the artist name from a referenced artist entity.
CoreData will not update song.artistName NSString just because you changed artist.artistName. It doesn't know that those 2 strings are related. You will need to do that update yourself.
Alternatively, you need to refresh the information from the Song entity to set the new variable of Song's artistName.
Hope that makes sense.
/* UPDATE */
Your Song references an Artist, but it is a to-many relationship. I always assumed a Song would only have one artist, so untick the to-many relationship in your CoreData relationship. In which case you should then display the artist name by way of song.artist.artistName. Then when the artist entity is updated the song will be referencing the artist which in turn has the correct info.