Thank you. I've got it working now, although I don't really understand why.
I followed your links, and compared the /apache/bin/php.ini files in my old and new XAMPP installations. The only setting that appeared relevant was '; For Unix only. You may supply arguments as well (default: "sendmail -t -i").
sendmail_path = "D:\xampp\sendmail\sendmail.exe -t"', which was commented OUT in the new installation. Deleting the ';' and restarting Apache solved the problem.
My computer is running WinXP, not Unix. I'm aware that Apache is normally run in the Unix environment, so perhaps that is the explanation. The main thing is, it works (and the e-mails went to my ISP and were routed back to my e-mail address as intended).
Isle of Mull, Scotland
OS=WinXP Home SP2