"Weights" actually refer to breech size:
A = 15/16"
B = 1"
C = 1 1/16"
D = 1 1/8"
The profile may be different between barrels of the same beech size - Rice, for example has different 45" barrels, the Allentown and the Southern Classic, and the Allentown has a smaller waist and tapers faster near the breech than does the Southern Classic. The Allentown will actually weigh less for a given "weight" than will the Southern Classic.
The A/B/C/D weight system worked well when pretty much all you had were the 37" Edward Marshall profile, the 38" Isaac Haines, and the standard 42" and 44" profiles that pretty much every company made with slight variations - you picked a length and a breech size, and that was it. People knew what you were talking about. Now that we have more profiles for a given barrel length, with significant differences in balance and actual weight for a given breech size, it has gotten a lot more confusing.