The truth is that the manufactures load the shells to match the game. And a 20 ga. w/ a mod. choke will likely put more shot in the rabbit than a 12 ga. w/ an improved cyl. choke.
I like to try and take several first time rabbit hunters to the woods / field each season and one of the things I like about rabbit hunting is that they can just bring any ole shotgun they have in the closet and it will do just fine.

Winchester Super X:
12 ga. 2 3/4" low brass #6 shot, velocity 1290, 1 oz. lead.
16 ga. 2 3/4" low brass #6 shot, velocity 1165, 1 oz. lead.
20 ga. 2 3/4" low brass #6 shot, velocity 1210, 7/8 oz. lead.
28 ga. 2 3/4" high brass #6 shot, velocity 1205, 1 oz. lead
410 3" high brass #6 shot, velocity 1100, 3/4 oz. lead