Barro's 2006 and 2009 rare disaster setup

Barro’s 2006 QJE rare disasters model has a different setup from his 2009 AER model. Specifically, in the QJE paper $v_{t+1} = 0$ (no disaster) has probability $e^{-p}$, whereas in the AER paper it has probability $1-p$. Both papers interpret $p$ as the probability of disaster per unit time, and they seem to reach similar conclusions about pricing. I realize that, for small $p$, $log(1-p) approx -p$, and therefore $e^{-p} approx 1-p$. Is this what he is implicitly doing? I’m unclear on why the two setups should be exactly the same.