GOOD NEWS! It wasn't the drive, it wasn't the cable, and it wasn't my PC! It was the NexStar 3 Enclosure!
I didn't have another PC with an eSATA port to test the enclosure, and I didn't have another eSATA device to test the PC, but I had a hunch it was the enclosure, so I bought a new enclosure (same NexStar 3 model), plugged it in, and it spun right up and Windows immediately recognized it!
This process of elimination took some time, but I'm glad my eSATA port on my 8100 Studio XPS is OK. The lesson here is that if you're having similar issues, don't discount the enclosure as the culprit, even if it still works through USB, because in my case, clearly the enclosure's eSATA mode was dead.
I hope I get more than 2 years out of the new enclosure!