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The fact that it works with polygons is there for backward compatibility. However, we recommend that you rather use points. You can do FeatureToPoint using the INSIDE option.
I presume, then, that the GPI is computing the polygons' centroids and using them for calculating the polynomial. Is that correct?
Yes you are correct.
Using the polygons as input would be the same as if you found the centriods using FeatureToPoint with the INSIDE option unchecked and then using that as input to GPI.
The problem with this approach is that often times the centriod falls outside the polygon, a letter C for example.