In a probabilistic seismic hazard assessment (PSHA) the estimated likelihood of rare/large ground-motions is controlled by the aleatory variability of the Ground-Motion Model (GMM). This variability is partly from the poor modelling of earthquake physics, and partly from the poor quality of GMM parameters. Here, we investigate the impact of the definition of moment magnitude (MW) on GMM variability. Indeed, MW can either be estimated directly from the moment-tensor inversion or deduced from other magnitude scales. However, for the same event, even the moment-tensor based MW estimates provided by different agencies often differ – due to differences in computation methods, inverted data, and network configuration. Various strategies have been proposed to define a unique MW value for each earthquake. In this study, we developed Fourier Spectrum GMMs using the pan-European Engineering Strong Motion database using two disctinct strategies to define MW: (1) the EMEC approach (ranking of the MW sources) preferring MW from event-specific studies, and (2) the IRSN approach (ranking + unification) in which all MW values are unified with those from the Global and Regional Centroïd Moment Tensor catalogs. We observe that the IRSN unified MW achieves a 18% smaller between-event variability at low frequencies (e.g. 0.3Hz) compared to EMEC MW. Such reduction in GMM variability makes a strong case for the use of MW unification approaches in GMMs and in earthquake catalogs – the two blocks of PSHA.