The two turbulence-generation models [Kaimal Spectrum Exponential Coherence (KSEC) and Mann] specified in the international standard for wind turbine design assume neutral atmospheric conditions and are based on statistical and spectral methods. Mainly due to the lack of physics, the flow fields simulated with these models ultimately differ in their underlying structure, especially in terms of the spatial coherence of longitudinal velocity perturbations. While this may not be critical for smaller wind turbine rotors, it becomes important when rotor sizes increase. Furthermore, it might be especially important in the context of floating technologies as they are more sensitive to large turbulent coherent structures. Previous work found that these differences between KSEC and Mann can propagate to loads predictions and thereby affect the design space of the entire wind turbine system. It is therefore crucial to determine in which ways these two models are underperforming. Up until now, validation of these models had only been done in the vertical direction because it is extremely difficult to obtain atmospheric turbulence measurements separated laterally, and sampled at heights relevant to wind energy. In this work, we address the lack of measurements by using high-fidelity, high-resolution simulation data as a reference. We perform hour-long, large-eddy simulations of turbulent velocity fields that are stability-dependent and contain three-dimensional coherent structures. These flow fields are then used to investigate which stochastic model is a better predictor of loads on a realistic spar-system floating offshore wind turbine, and to quantify how the assumption of neutral stratification propagates to short-term load estimates. Both stochastic turbulence models are found to overpredict fatigue loading in high-wind scenarios (in some cases, by more than 25%) and underpredict it when the wind speed is low (by as much as 20%). The KSEC model matches the high-fidelity flow fields more closely than Mann at high wind speeds, and the opposite is true at low wind speeds. Finally, turbine loading is found to be sensitive to atmospheric stability even when the turbulence intensity remains fairly constant. This sensitivity is most pronounced at low wind speeds, when fatigue load estimates on the spar system can differ by 40%.