In 102100 and 3857, we can use an ellipsoid/spheroid-based geographic coordinate system because the map projection algorithm is handling the conversion to a sphere internally.
102100: PROJECTION["Mercator_Auxiliary_Sphere"]
102113: PROJECTION["Mercator"]
This parameter: PARAMETER["Auxiliary_Sphere_Type",0.0]
determines how an ellipsoid-based GCS should be handled in a map projection that supports spheres only. The zero value means to use the semimajor axis for the sphere's radius. "1": use semiminor axis, "2": calculate authalic sphere's radius, "3": calculate authalic sphere's radius and convert geodetic latitudes to authalic latitudes.
With 102113 (3785), because the GCS was sphere-based, technically you would have to do a geographic (datum) transformation from WGS 1984 to the sphere-based GCS before the projection could occur. This is more difficult to handle because it means all data has to go through a geographic transformation. In 102100 (3857) only non-WGS84 data has to go through a geographic transformation.
Both definitions give exactly the same results.
Melita
(edited to add sentence about results, 1:22pm PDT)