Price specifically stated that the injury was in the muscle, not in the ligament. Therefore, its a strain, not a sprain, assuming he is telling the truth and assuming Andrews got the diagnosis right. If it was the Red Sox team physicians who drew that conclusion I would be more skeptical, but Andrews has a very good reputation. You know as well as I do that strains may cause localized inflammation and some edema. The MRI would likely have been able to pick up a torn or partially torn UCL despite the swelling, so I would interpret "inconclusive" to mean "no tear in the ligament seen". When you think about it, nearly every injury that requires an MRI is going to have some swelling associated with it.
And thats good news if you are a Red Sox fan. I admit that there is a small chance the the MRI missed a tear, but the resolution on that test is so good that I would imagine that the results are probably over 90% accurate. As you said, time will tell.