To play the required record you indeed need to play the exact song the item is requesting. Hints are given in the phrases when looking at them. So you actually need to have that song on your computer, so that you can pick it up in the explorer! Yes, it's quite a strange mechanic, in fact I'm thinking to let the user pick any song by one specific artist for each item, instead of the exact song.
Let me know if you figured out some of the songs!