Many thanks for reviewing the model.
I have modified the model and its description since. Please take another look, especially the section on Super-Gaussian and choices of parameters and their consequences.
I think the formulation used in the model may be correct and there is no inconsistency.
The regular 1-D Gaussian (1-D because the support is a line) applies to the axial cross section of the beam. 2.35 ≈ 2×1.1774 comes from the FWHM being full width on both sides of the axis. The area under such flat PDF curve between FWHM limits would be some 76% of the total. This has nothing to do with beam dimensionality.
If you rotate such distribution around its axis, that becomes 2-D isotropic Gaussian (now the support is a plane). I think that one of the fancy names for it is isotropic bivariate Gaussian/Normal with equal variance and zero covariance.
The angular intensity distribution is still 1-D Gaussian as before, but now to get the portion of the flux contained in the beam cone (or greater than half max intensity), you need to find the volume under the surface, not the area under the curve.
For the rotated Gaussian this volume has a nice property: For FWHM limits, 50% of the volume (flux) is in the hotspot and 50% outside it. For more general FWpM limits 1-p fraction of the flux is contained within the hotspot. This does not hold for Super-Gaussian and you need to integrate to get it. The model description has a bit more on that.