The ab initio calculations were correlated with magnetic and emission characteristics to understand the modulation of properties of NIR-emissive [YbIII(2,2'-bipyridine-1,1'-dioxide)4]3+ single-molecule magnets by cyanido/thiocyanidometallate counterions, [AgI(CN)2]- (1), [AuI(SCN)2]- (2), [CdII(CN)4]2-/[CdII2(CN)7]3- (3), and [MIII(CN)6]3- [MIII = Co (4), Ir (5), Fe (6), Cr (7)]. Theoretical studies indicate easy-axis-type ground doublets for all YbIII centers. They differ in the magnetic axiality; however, transversal g-tensor components are always large enough to explain the lack of zero-dc-field relaxation. The excited doublets lie more than 120 cm-1 above the ground one for all YbIII centers. It was confirmed by high-resolution emission spectra reproduced from the ab initio calculations that give reliable insight into energies and oscillator strengths of optical transitions. These findings indicate the dominance of Raman relaxation with the power n varying from 2.93(4) to 6.9(2) in the 4-3-5-1-2 series. This trend partially follows the magnetic axiality, being deeper correlated with the phonon modes schemes of (thio)cyanido matrices.