Right. I think I've cracked it. For the Mac, anyway. Audacity doesn't seem to have the capacity on its Mac download to record what's playing through the computer, and Audio Hijack, as far as I can make out, is no longer free.

But Audacity recommends either using a loopback cable, to connect your line-out to your line-in, or using a free download called Soundflower - after pressing the Free Download button, go to Featured>Downloads>Show All in the left-hand column to decide which is best for your system. I've installed it, and it seems to work fine so far.