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.