I tried Mark's suggestion, which worked. When I searched my entire Cake installation for "Configure::write('debug'", I found tons of lines to turn debugging on - although most of these were present in the original files as downloaded from the official CakePHP site. As an experiment, I made a copy of my Cake installation, changed all instances that set debug higher than 0, and that worked. Then, assuming that some of them should be there, and not being sure which were supposed to be there, I decided to completely delete CakePHP, download a fresh copy, and replace it.
After I configured the new core.php, the missing controller error was replaced with a not found error, and the parse error that I intentionally caused for testing purposes was replaced with an internal server error. I'm not sure that showing an internal server error is what it is supposed to do... but the behavior did change after editing the debug setting, and it doesn't reveal the file paths anymore. So, I guess that it's behaving the way that it's supposed to.
I would like to give Mark credit for helping me solve this, but he only posted a comment.