Generalizing the mode-coupling theory for ideal liquid-glass transitions, equations of motion are derived for the correlation functions describing the glassy dynamics of a diatomic probe molecule immersed in a simple glass-forming system. The molecule is described in the interaction-site representation and the equations are solved for a dumbbell molecule consisting of two fused hard spheres in a hard-sphere system. The results for the molecule's arrested position in the glass state and the reorientational correlators for angular-momentum index l=1 and l=2 near the glass transition are compared with those obtained previously within a theory based on a tensor-density description of the molecule in order to demonstrate that the two approaches yield equivalent results. For strongly hindered reorientational motion, the dipole-relaxation spectra for the alpha process can be mapped on the dielectric-loss spectra of glycerol if a rescaling is performed according to a suggestion by Dixon et al. [Phys. Rev. Lett. 65, 1108 (1990)]. It is demonstrated that the glassy dynamics is independent of the molecule's inertia parameters.