Thanks Fred28, dragging out of the Applications folder didn't work, but using the Finder did. So I installed a new version of WavePad and tested it. If I record, stop the recording, and (even if I click at the end of the recorded material) start recording again, the timer will revert to 0, erasing what has been recorded. I can reliably reproduce this.
But I saw I can insert a new recording if I click on a specific location inside the recording and then record. I tested this to make sure it was consistent. Also I found that the magic Undo button will restore a recording that gets deleted, although I hate to rely on this. I will say that the "Are you sure you want to stop recording?" dialog box makes me save the file immediately. And of course, by now I use the pause button as much as I can.