I don't know about a grading system per si... But, generally the more expensive usually means, the more ornate and better finished the barrels are. And, of course you have different thicknesses and tapers too.
Now I am in the process of figuring out from reading various articles about modifications done to Axis's, that most of the barrels from Savage firearms you mentioned are interchangeable, and can be mounted on the Axis Receiver (by a competent Gunsmith) providing they use the Savage barrel nut mounting system. Also the Bolt Head is interchangeable with the Savage 110 series as well, and maybe others. The Axis will also except the "AccuTrigger" on them, and a couple of manufactures make aftermarket triggers for them too. The Stock still seems to be the weakest link as far as choices to improve or mod in the aftermarket selection.
Now back to the original question... I'm not sure if the way Savage makes the barrels on the inside (Rifling and Bore) is any different on the "low cost" Axis type barrels v.s. the "Higher cost" other barrels or not. Or, if they use the same rifling tools, and machines or not... I do know that my wife's Axis in .223 is a shooter, and I'm hopeful that once I can hit the range with my (Left Hand) Axis in .223 it will be a shooter too.