Rex, you're completely correct. Thanks for adding the all-too-important details that I neglected. The beam comes in from the left, from 180 to 0, and the data have been normalized to the 90-degree value. The energies and spatial distribution go hand-in-hand in that the forward beam of neutrons will also have higher energies.
These distributions are for a beam impinging on a stationary, thin target, i.e. the region of neutron formation has zero thickness (order of 1-5um usually). The deuterons quickly lose energy due to the target, and you thus get fusions at energies lower than the incident beam and distributed by depth. I believe the energy range (not the depth) is what leads to the higher-order terms than (cos(theta))^2.
If you have no energy loss, the neutron distribution would be purely a (cos(theta))^2 distribution due to spin-orbit coupling (due itself to the deuteron center-of-mass and center-of-charge offset) which gives rise to p-wave scattering (see
https://ir.library.ontariotechu.ca/bits ... sequence=3 and
https://link.springer.com/chapter/10.10 ... -45878-1_6). For those curious, this is because in scattering theory, d(sigma)/d(Omega) = |f|^2, where the p-wave term of f is f ~ P_1(cos(theta)). Here sigma is the cross-section, Omega is the solid angle, and P_1 is the 1st Legendre polynomial P_1(x) = 2x-1. The cosine argument gets squared, which is where the distribution arises. Integrate the neutron emission for each "slice" of the target, taking into account the highly nonlinear stopping power of ions in matter (Bragg peak) and I think you'd get the coefficients in that paper. The fact that the A2 terms in table 2 are the largest in magnitude for their rows supports this idea. There's a chance I'm wrong, but it seems to be a pretty solid explanation.
I suspect these measurements were made using a D+ beam on a pre-loaded titanium or zirconium target. The beam spot size was likely on the order of a few mm to cm, as is the case for most such systems since it helps distribute the heat load and reduce target sputtering. I do not know if these data reflect that finite size, or if they have been altered to represent a point source. My guess is that the measurements were all performed in the far-field, and hence the finite size makes no difference.
A fusor does have areas that behave like thin targets, such as the cathode and endcaps. Beam-background gas fusion is technically not a thin target as the range of D+ and D2+ in D2 gas is on the order of cm, but like with the thin target, one could theoretically integrate the distribution along the appropriate paths to generate the distribution. This would be very complex, as it would require you to have an idea of the spatially-varying ion energy distribution.