Image of shape under Möbius transformation

Your method is mostly sound. The one improvement I can suggest is to think of $\infty$ as the limiting value as $|z|$ becomes large in any direction; that is, in the (extended) complex plane there is no difference between $\infty$ and $-\infty$. Therefore a "circle" through $\infty$ is just a line, but its direction is determined by the other two points, not by the "sign/phase of $\infty$".

Three points on the circle are $0$, $-1+i$, and $2i$, which are mapped respectively to $\infty$, $i$, and $0$. Therefore the image is the "circle" through those three points, which is the (vertical) line through $i$ and $0$.