These definitions are consistent. In the 2-state model, mu is the specific transition rate from the alive state to the dead state, and the q probability in the limit is the probability of moving from alive to dead in a short period. Considering the general formula for the transition rate for a Markov jump process, then we have to set i = A(live), and j = D(ead). So pij(h) = pAD(h), which is the probability of moving from alive to dead in a short period (just like the q probability in the first definition). And pij(0) = pAD(0) = 0 (as you cannot move from alive to dead in no time). So the two formulae are saying the same thing.
