Cut rifling and Button rifling refer to how the rifling was created in the barrel. Cut rifling is made by individually cutting the grooves, one at a time a little at a time. Cut rifling is supposed to impart little to no stress on the barrel.
Button rifling is formed by pulling a slightly oversize carbide 'button' through the bore, which presses the grooves into the steel. Button rifling is supposed to be a little 'stressful' on barrels, but I'm not sure how much.
Freebore is really just an extra long throat, and not a rifling type. Weatherby uses it to keep the pressure from peaking so steeply on their magnum rounds.
Polygonal rifling is primarily used in handgun barrels. It can only be used with jacketed bullets, as cast bullets lead the barrel very fast. It is however, supposed to give a slight velocity increase with jacketed bullets. I don't know of anyone who uses Polygonal rifling in rifle barrels, but that doesn't mean that no one does.
As far as one type rifling stabilizing a bullet better than another, I don't think it would. That would depend more on the rate of twist that your barrel has. A faster twist equals a more stable bullet(more or less).
Hopefully this answered some of your questions. If not, maybe one of the serious shooters will chime in, and set us both straight.