In my opinion, geometric surface normal is the cross product of triangle edge vectors, and shading surface normal is the interpolation of the predefined normals at the three vertexes.
I used Mitsuba to render the geometric normal channel and the shading normal channel:
geometric normal:
shading normal:
Your existing opinion is correct, though there's one extra detail. The geometric normal is the normal of the actual triangle, based on the vertices' positions (the cross-product of edge vectors, as you said). The shading normal is altered from this, and is used when shading a fragment or ray hit. In the simple case, it's the interpolation of the three normals set at the vertices. This is used for curved objects to hide the polygon boundaries. It might also be modified by a bump map or normal map, to add finer detail to the appearance than the polygon mesh has.
