We present HST WFPC2 observations, supplemented by ground-based Halpha data, of the star-cluster populations in two pairs of interacting galaxies selected for being in very different kinds of encounters seen at different stages. Dynamical information and n-body simulations provide the details of encounter geometry, mass ratio, and timing. In NGC 5752/4 we are seeing a weak encounter, well past closest approach, after about 2.5 x 10(8) yr. The large spiral NGC 5754 has a normal population of disk clusters, while the fainter companion NGC 5752 exhibits a rich population of luminous clusters with a flatter luminosity function. The strong, ongoing encounter in NGC 6621/2, seen about 1.0 x 10(8) yr past closest approach between roughly equal-mass galaxies, has produced an extensive population of luminous clusters, particularly young and luminous in a small region between the two nuclei. This region is dynamically interesting, with such a strong perturbation in the velocity field that the rotation curve reverses sign. From these results, in comparison with other strongly interacting systems discussed in the literature, cluster formation requires a threshold level of perturbation, with stage of the interaction a less important factor. The location of the most active star formation in NGC 6621/2 draws attention to a possible role for the Toomre stability threshold in shaping star formation in interacting galaxies. The rich cluster populations in NGC 5752 and NGC 6621 show that direct contact between gas-rich galaxy disks is not a requirement to form luminous clusters and that they can be triggered by processes happening within a single galaxy disk ( albeit triggered by external perturbations).