Hi Jim and welcome to tom's hardware!
Basically this shouldn't work. More often than not, you can't just take a hard drive with Windows installed on it from one computer and put it in another PC and boot to Windows properly. This wouldn't work unless the motherboards of the two computers are the exact same model or at least pretty similar models.
As for the hard drive troubles you're having, I'd suggest that you try the HDD with a different SATA port and cables, to see if the same thing happens. It seems like it's working fine with different computers, so the issues should be related to the computer that can't detect it. If the different port/cables option doesn't prove helpful, you could try updating your BIOS/UEFI, to see if the HDD is detected afterwards.
However, if none of the above proves useful, I'd suggest that you contact the computer manufacturer's customer support and ask for further assistance. Some OEM computers require you to get specific hardware in order for the device to work properly, so that might be the cause for this problem as well.
Please let me know how everything goes.
Boogieman_WD