Monoatomic X- (X = O, S) chalcogen centers in MZ (M = Na, K, Rb and Z = Cl, Br, I) alkali halide lattices are investigated within the framework of density functional theory with the principal aim to establish defect models. In electron paramagnetic resonance (EPR) experiments, X- defects with tetragonal, orthorhombic, and monoclinic g-tensor symmetry have been observed. In this paper, models in which X- replaces a single halide ion, with a next nearest neighbor and a nearest neighbor halide vacancy, are validated for the X- centers with tetragonal and orthorhombic symmetry, respectively. As such defect models are extended, the ability to reproduce experimental data is a stringent test for various computational approaches. Cluster in vacuo and embedded cluster schemes are used to calculate energy and EPR parameters for the two vacancy configurations. The final assignment of a defect structure is based on the qualitative and quantitative reproduction of experimental g and (super)hyperfine tensors.