Okay! I also tried it on three Ubuntu machines, and there I got the full path in root cron. But maybe this is different between pure Debian and Ubuntu?
The weird thing is I had a backup script running in root cron for weeks. And these backups always worked. Suddenly I get email notifications from the cron daemon about this problem. So for one reason or another, in the past I had the full path. Otherwise the backup script could have never worked like it did before.
So that will always be mystery, but let’s close this issue then. Thanks for the excellent support in any case!