I thought I understood what happened here when I asked this question yesterday. However now I'm not so sure anymore. My explanation was: The first parametrization uses $[-\pi,0]$ which happens to be included in the domain of the principal branch of the logarithm and thus the corresponding integral yields something different than the one using $[\pi,2\pi]$ which is included in the domain of a side branch (if you call it like that in english). But actually the square root must be taken of $\gamma$ which in both cases (once you plugged t into the exponential function) is the lower arc of the unit circle, so we shouldn't have two different domains yielding two different branches.
Long story short: Where exactly in those two integrals is the point the different branch cuts step in? I just can't see it.