To me, it sounds almost as if the (presumably) IDE cable between the MoBo and the CD drive isn't connected properly - or it could be that the pins on the back aren't configured correctly to work in the other computer...
The drive itself will (in my experience) always put the light on when the drawer is closed, this is it doing a check to see if there is anything in it - if there is then the OS would (normally) check to see if there is an "autorun" type of file there to continue - or alternatively, if it is a music CD and as long as the OS is configured to, start playing the audio...
I would check the cable is fully connected at both ends, so I would remove each end (one at a time) and push it firmly back in again. I would also check what the pins are set to (master, slave, cable select) on the original drive, then set the same configuration to the alternate drive. Bear in mind that different drive manufacturers use different combinations, so you should be able to see some sort of a diagram on a label somewhere on the drive case.......
Let us know how it goes....
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I've seen the light... It was green, flashy and attached to a Network Interface Card...
Whenever someone says "You can't miss it" I invariably do...