The standars explanation is nice, but only a conjecture. But I believe it. Why not? Until someone comes up with a better theory.No historycan tell you the answer. Nobody made a note of what they meant when they wrote it.(Mind you, the bit of paper might just be lurking in that old box in your attic?).
By the by, the Great Plague of 1665 seems a lot more likely to methan the Black Death of the Middle Ages. I cant really see a rhyme lasting in that form as long as that. And the verse structure seems subjectively to me to have more 1660 than 1360 about it. (Having said that, "Summer is icumen in" sounds dead modern but is definitely medieval).
Agreed, we need Masato or Malcolm Douglas to chase up the earliest known date first, before getting carried away.