Something that's worked very well for me:
A very easy route for the steels mentioned (420HC, CV, etc), for touching up on a stone before stropping, is using the Fine side of an oiled India stone (~360-grit, give or take) to make a few light edge-leading passes; enough to form a very light & thin burr again and no more. The stone works real well on these types of steels, removing enough metal quickly enough, that it'll be immediately responsive to stropping of most any kind, with a compound or not. My own stropping preference after using this stone is just doing so on a bare piece of paper laid over the oiled stone. But if a compounded strop is the preference, green on leather will also work well when used with a light, brushing pass or a few (no more than maybe 5 - 10 passes per side). When I was using green compound, I used it on the sueded side of a leather belt. It was just right for getting that hair-popping sharpness with a minimum of work or fuss, AFTER making sure the edge was fully apexed (verified by the burr) on a stone. If preferring an edge with some toothy bite on these steels, don't take it too far with the green compound, as it'll quickly polish away a lot of that toothy bite. If you want a more completely polished, true shaving edge, I'd suggest stropping with the green on something firmer than leather, like a firm & tight-grained wood. That will allow you to spend more time on the polishing without risking rounding off the apex, as would likely happen on a leather strop used the same way for an extended period of time. And the firmer wood substrate will also allow the compound to work faster to a high polish, than if used on something softer.
Regarding the India stone, I've also used a pocket-sized version (4" x 1") of the Fine India for very quick, on-the-go, tuning up of these steels. I also use this with a little bit of mineral oil, as that keeps the stone unclogged & cutting efficiently with a light touch, and keeps it clean with just a wipe-off using a microfiber towel. 420HC, CV and similar steels are a perfect match for a stone of this type. I carry either or both of Buck & Case folders in these steels (420HC, 425M, CV) on a daily basis, so I'm a true believer in how well this works.
For the 8Cr13MoV, I'd start leaning toward using an aluminum oxide polishing compound like white rouge, Flitz, Simichrome or similar compounds. This steel is a little more wear-resistant than the others mentioned; green could still do OK, but aluminum oxide will do it better (quicker, more effortlessly to high polish). Same thumbrules apply, with regard to making sure the edge is ready for stropping, with appropriate stone work first.
