BSODS are almost always hardware related either the hardware itself or a corrupt driver. The fact that you have tried the HDD in a similar machine, although not conclusive as its a different machine and therefore not matched to that machine, would suggest that the Windows update has caused corruption to a driver.
This is a puzzle as automatic update doesnt normally update drivers as they are listed as optional.
On the clean HDD boot up and turn off automatic updates this will enable you to install updates individually.
You could try starting up in safe mode as this loads a basic set of drivers so that hopefully you can check which updates have been applied and back them out one at a time.
If this doesnt work you can download updates from the Windows Update Catalogue one at a time and apply them until the problem occurs.
Alternatively apply the latest updates to a machine where they dont cause the problem then you can identify which updates have been applied and then do a web search on the updates ID to see what the update contains and/or download them from the catalogue and apply them one at a time.