Confused about the shape of 3D s-domain plots
A "pole" is defined mathematically by an equation whose denominator becomes zero at a particular or several values of s; where s = \$\sigma +j\omega\$. A particular example is a 2nd order function like this: -



This happens to be a low pass filter and, if you set the denominator to zero and solved it you would find one or two values for s that tell you where the response is infinite. At all other values of s the response is non infinite and clearly forms the peaking shape in your question.

The shape has nothing to do with the driving waveform (probing waveform?)

Here's a picture that might help: -



It assumes a 2nd order low pass filter and has solutions for s shown in the bottom right picture.