The attractive-regime (pulling) constant-height manipulation of a ${\mathrm{C}}_{60}$ molecule on the Si(001) surface is modeled using density-functional theory with a scanning tunneling microscope tip included explicitly in the calculations. We demonstrate that the structure of the tip and its position with respect to the ${\mathrm{C}}_{60}$ prior to the pulling manipulation process determine its outcome. No translation of the molecule was achieved for some tip trajectories, while for others, the molecule was successfully translated via the pivoting mechanism reported previously [D. L. Keeling et al. Phys. Rev. Lett. 94, 146104 (2005)]. We also find evidence of possible transition between different manipulation modes: the initial mode of manipulation may not necessarily be preserved over the whole tip trajectory, e.g., the pushing mode may go over into the pulling one. Our results agree with the experimentally found relatively low success rate of pulling manipulation and underscore the role of the tip in pulling. The calculated tip forces have a sawtooth pattern that is correlated with the main bond-breaking and bond formation events during the manipulation.