PSD in MATLAB, problem with the coefficient $\frac{1}{2\pi}$
I would simply say that it doesn't matter so much. The Fourier Transformation and its inverse are a pair and the two formulas are entangled and require a normalization factor, which is up to conventions. See eg enter link description here

If your theory/model does not require a special convention, it is basically a degree of freedom that you can choose to fit your needs.