Single-ion magnets 1 and 2 and their diamagnetic analogues 3 and 4 for magnetic-site dilution were obtained through substitution of the coordinated water molecules of [Ln(TTA)3(H2O)2] (Ln=Dy (1, 2), Y (3, 4); TTA=4,4,4-trifluoro-1-(2-thienyl)-1,3-butanedionate) by 2,2′-bipyridine (1, 3) and 1,10-phenanthroline (2, 4) capping ligands. Their structures and magnetic properties were investigated with the goal of identifying features relevant to modulating relaxation dynamics of single-ion magnets. The metal ions in all complexes adopt an approximately square-antiprismatic (SAP) O6N2 coordination environment. The SAP polyhedrons for both 1 and 2 show slight longitudinal compression, while the coordination sphere of 1 deviates more from an ideal SAP than that of 2, as indicated by the skew angles of the SAP environment. The similar values of Ueff for the two magnetically diluted samples imply nearly the same distribution of low-lying states for their DyIII centers, which is consistent with the slight axial contraction observed for 1 and 2 and further corroborated by ligand-field analysis. The fast quantum tunneling rate τQTM of 1, which is about ten times faster than that of 2, can presumably be associated with the larger rotation of the SAP surroundings. This distortion may result in a significant transverse anisotropy terms, and thus strongly affect the dynamic behavior of the system.