Pretty sure the other vertices are coming from the UV map. You can tell this isn’t hidden or new geometry because in both screenshots you have 960 triangles.
According to image #2, you have (2) UV channels on this sphere. Unless you specifically un-checked the option “Generate Mapping Coords” for this primitive in Max, then yes you do - whether you add one or not. Triangulating quads increases edges, not verts btw - but y’all seem to have it figured out.