To my knowledge there is no SAAMI specification for anything like .44 Magnum+P+ or +P for that matter. That rating is just a marketing gimmick used by the manufacturer.
Did you check out the specs on the round I mentioned as a +P+ round? Maybe it isn't official but clearly it's a round that is very powerful compared to the original .44 magnum specs. Here are those spec for the Buffalo Bore round they are calling +P+.
340 gr. 1,478 fps/M.E. 1,649 ft. lbs.
That may not have official specs making it a true +P+ round but it dang sure is a powerful round for a .44 mag.. I'm looking at a .50 cal. cartridge with specs like this.
325 gr. 1450 FPS
So the .44 mag. round labelled as +P+ is likely more powerful than this .50 cal round. It doesn't have the ft./lbs. listed but I would have to assume a 325 gr. bullet going slower than a 340 gr. bullet is going to have less energy. Either way it's a big load for a .44 mag. and if Buffalo Bore, which has a very good reputation, says not to use it in a S&W then I'm not going to. I know that early models of the 629 had issues with 300 gr. bullets. They would jump time and all sorts of things like having the cylinder spin backwards because of shooting rounds that were too powerful. Again the Endurance Package helped deal with that issue but not to the point of firing a 340 gr. bullet 1478 fps. That's a lot whether it's officially a +P+ or not. Maybe there's marketing involved but there's also physics involved. That round is too powerful for S&W revolvers from all I've ever heard.