You might not be able to, but a mammalogist, or a reindeer herder, probably could. There are a number of subspecies of reindeer/caribou, though I am not sure how distinct they are (and I had not been aware of the possible interbreeding between domestic reindeer and Alaskan caribou that yabob mentions). At any rate, I would expect that someone knowledgeable most likely could tell them apart.
Come to think of it, were common names even common at that time? I can well believe that e.g. a blue tit or a red-tailed kite would have different names in different parts of England at the time.
They certainly weren’t standardized, or “official.” Even today there are no official global standards for common names.