I recognize a difference among honing, lapping, and polishing. I don't know exactly how each of these are applied throughout the industry (there is some ambiguity or overlap), but I think the distinctions are generally recognized.
I would never "hone" one of my barrels. I don't have the equipment to do that, and to me "honing" implies removing more metal than I'm ever interested in doing. This may be "just semantics", but semantics can be important.
What I use for lapping or polishing (which I personally regard as the next two steps, in order of effect on the surface) are the
Timesaver Lapping Compounds (picked up at an old shop auction some years ago). It's generally critical for me to use NON-EMBEDDING compounds, and both the green (steel) and the yellow (brass/bronze/soft metal) Timesaver compounds are non-embeding. As directed, I just mix up a paste with the desired compound and oil, and then (for my barrel), simply put it on a patch and run it up and down the barrel a bunch of times. I'll use a couple of the grades (usually "fine" and "very fine", and on occasion a quick pass with "medium") since the stuff I have is not heavily abrasive. Since these are non-embedding, cleaning them off/out is not all that much of a chore, although it does require some repetition.
I also have used these fairly extensively on brass, but that's another story.