You are awesome professor Dave.Close, but not quite. You have most of the pieces, but not assembled correctly.
What are the real and imaginary parts of ex(t)+iy(t), if x(t)=2πcost and y(t)=2πsint?

I know that parametrization is very basic, but the way how you noticed that the annulus sketch of the OP was wrong is super. I have always thought that transforming circles under ez gives annulus with inner radius e−r and outer radius er where r is the radius of the disk.
This makes me wondering what the heck the analysis of e−2π≤∣w∣≤e2π even means????!!!
Professor Steven example in the OP was so beautiful and it showed how may someone get wrong easily when he thinks that he was always right. I hope that professor Steven caught his error and thanks a lot Dr.Peterson.