Grumble wrote:
snip
I'm hoping this is a problem with the PSU, not with the HDD (I'd say
that last experiment would point to a failing PSU, no?)


That seems likely to me.

What are the dangers of plugging the data connector with the power
turned on?

Don't do that. Most drives aren't designed for it (unless you've got
an IDE drive spec'd for hot-plugging or a SCSI drive, which often are)
and you can blow out the electronics on the drive by doing it.

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