using knoppix tohd=/dev/sda1 boot option when sda1 is an external USB HD

Hello carolus!

I would suggest that you copy the directory KNOPPIX and its subdirectories from your Knoppix CD to the root directory of your USB HDD. Then use a bootonly CD while the USB HDD is connected to your old computer and watch whether KNOPPIX will be found on that HDD. As long as there is no overlay file in the KNOPPIX directory you will be asked to create one.

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In contrast, on my target machine, with only an external USB hard drive, ..

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This machine has no internal hard disk, only an external USB drive?

You didn't tell us the output of "blkid" and "fdisk -l" and you didn't tell us the error messages while trying to boot with the cheatcode "tohd=/dev/sda1"