Computation of contour integral without winding numbers.

I am trying to solve problem $$3$$ on page 108 in Ahlfors. The problem asks to compute $$\int_{\left\vert z \right\vert = 2} \frac{dz}{z^2 – 1}$$ so I am trying to do so without the use of winding numbers, which isn’t introduced until later sections.

I first used partial fraction decomposition to write $$\frac{1}{z^2 – 1}$$ as $$\frac{1}{2} \cdot (\frac{1}{z-1} – \frac{1}{z+1})$$. From here, it’s clear that the integral is $$0$$ if we resort to winding numbers. Since I was unable to compute the integral using elementary methods, I resorted to reading this solution. However, I don’t understand how they changed the path of integration from $$\left\vert z \right\vert = 2$$ to $$\left\vert z – 1 \right\vert = 1$$ because when $$z = -2$$, $$\left\vert z – 1 \right\vert = \left\vert -3 \right\vert = 3 \ne 1$$.