They are both correct. :-)
I gave a somewhat detailed write-up last year on my blog, but the gist of the argument is that if $\tau = 0$, then $d\tau = 0$. On the flip side, if $d\tau = 0$, locally you can lift $\tau = du$ for some function $u$, and by replacing $\omega$ with $e^{-u}\omega$ you construct the desired equi-affine volume form.
In other words, the second statement applies to arbitrary volume forms, while the first statement that $\tau$ vanishes applies to a special volume form, the one that gives the equi-affine condition.
Just to make clear that there's no contradiction. We first note that for any volume form (non-vanishing differential form of top degree), the expression $\nabla_X \omega = \tau(X)\omega$ holds for some one-form $\tau$ depending on $\omega$. Since we are interested in local statements, assume now our manifold is diffeomorphic to the unit ball.
The first statement says that "$\exists \omega$ a volume form such that $\tau = 0$."
The second statement say that "$\forall \omega$ a volume form, the corresponding $d\tau = 0$."
The two statements are equivalent using what I sketched in the second paragraph above.