I've been a musician for over forty years now. Despite never being a payed studio musician I've know people where were. Here's how it works. The studio needs a version of a song. Because the musicians are employed by the studio everything they do is owned by that studio. So you will never see who actually performed or wrote that version of that song. That's why the song 'That old black magic' is listed as the song played during the closing credits without any changes made to who did the music. So in effect the movie studio bought rights to use that song from the owners, the recording studio. If there are two versions of this song in the movie then the movie studio's employees created a second version. With no credit going to them. In the end the movie studio paid once for the rights to use it. And made three versions of it to save money.
How I figured it out. I sped up the playback of the end credits. Then comparing it to the bridge of the song. I transposed the pitch in my head to get a match. Even heavily stylized as it is there is enough there for me to hear it.