Electronic – this antenna called, and what does the polarization and radiation pattern look like

For a compound box antenna the formula is similar and this might be useful: -

Obviously they are not using A and B to denote the aperture dimensions but, I suspect that "r" is going to be exactly the same i.e. distance between centre of aperture and measurement point.

The inclusion of r makes it a near-field measurement. The far-field measurement is an approximation based on the assumption that r is approaching infinity, which in practise means that r is many times larger than the antenna.

If you care about near-field effects, then you must use the near-field form. If you don't care about near-field, i.e. if your antennas are spaced apart by at least a few tens of their size, then you should simplify your report by using only the far-field radiation patterns.

The near-field expression probably gives you a direct expression for field-strength or power at each (r, theta, phi) point in space, whereas the far-field expression usually just tells you the antenna gain with respect to isotropic (dBi) in each (phi, theta) direction. The way you use each expression therefore differs because one includes propagation loss and the other does not; you must account for 1/r^2 (= -20log(r) dB) propagation losses if you're using the far-field patterns to compute total power transfer between antennae.