Oh, and there's some canon stuff to support this theory, too.
First, take a look at a "normal" Klingon - specifically, the Klingon Ambassador in the 2280's/2290's:
Now, here's a "Discovery" Klingon - specifically, T'Kumva:
Now, here's the Klingon captain from "Into Darkness":
If you notice, the Klingon captain's features are somewhere between the other two, suggesting that he might be a hybrid of the two different Klingon subspecies. And while "Into Darkness" DOES take place in the Kelvin universe, as opposed to the Prime, the Klingon captain's age would suggest that he was born BEFORE the two realities diverged.