JAK1 and JAK2 do appear to mediate signaling by different receptors, but there is overlap. In other words, some work through one or the other, others work through both.
JAK2 signaling appears to be a bit more discrete over JAK1. The wikipedia articles give a nice quick overview if you're not inclined to dig super deep. Usually these types of mechanisms are very sensitive to cellular context.
As for EXEL's claims, I can't dispute them. But at this point, claiming importance for JAK2 specificity doesn't seem all that big a deal if you don't have clinical data to show that it matters. Know what I'm saying?