I (straightforwardly) got the same generating function as you, then spent a while thinking about it before deciding to finally look up the definition of "composition." Your generating function is right if p(n,m) is the number of *partitions* with pieces of size <= m. But in a composition, the arrangement of the terms matter (look it up on Wikipedia).
I've basically figured out the problem now, which I got by working out a recurrence relation for p(n,m); let me know if you're still stuck with the new and improved definition.