Flitz IS abrasive (aluminum oxide at about ~3 microns, per the manufacturer). So is Simichrome (aluminum oxide at about ~9 microns). Whatever 'staining' of wood handles might happen with these, is more likely due to metal swarf (black residue) being rubbed into the wood. Either of these are excellent for polishing steel blades. If protecting wood handles is a concern, perhaps taping or otherwise covering them might be the simplest solution.
If using wet/dry sandpaper to remove fine scratches, always start high in grit, only going lower if it's not effective. If sanding a satin-finish blade, anything in the 400-800 grit range works pretty well. If a polished blade, start as high as possible, at 2000+. Those might even leave some haze on a mirrored blade, but cleaning that up with Flitz/Simichrome is not too difficult, at that level of polish.
The 'white diamond' metal polish might be tin oxide. It's abrasive, but much softer than aluminum oxide, and wouldn't likely do much by hand on a hardened steel blade. Might be somewhat effective on a powered/driven device, like a buffing wheel. For what it's worth, any real metal 'polish' is likely to be somewhat abrasive, else it wouldn't polish at all. The products called 'non-abrasive' are usually soft enough to avoid damaging metals for which they're intended, but still abrasive nevertheless. The gentlest ones would be those recommended or designed for use on plated jewelry or antiques (gold/silver plating). If too abrasive, they'd strip the plating off (Flitz/Simichrome would damage them).
David