In my experience, honing and sharpening have been used to describe the same thing, although I think that maybe honing has evolved to mean the final stages of sharpening.
I think of burnishing to mean a process which treats the surface of steel by using the malleability of the steel to smooth out surface scratches and burrs as a means of polishing or refining an edge. The “steels” I’ve used have been more like longitudinal files than burnishers, but I’m sure that some steels could be burnishers. I think of stropping as a burnishing process. Take a rough piece of steel to a powered wire wheel and you’ll see the effects of burnishing.