Much of the bullet diameter conflict stems from old 32-20s vs newer 32-20s. The older revolvers and rifles were nominally 311 but could be found up to the area of 313/314.
When the t/c Contender was introduced t/c opted for the more common .308 diameter thus sometimes it is called 30-20. Ruger chambered a revolver that used 32-20 and some had a second cylinder for 32 H/R. Marlin also offers a rifle in 32-20. In using standard velocity and lead bullets, the larger bullets would swage to the .308 barrel. The trouble begins in trying .308 bullets in .311 and larger bores. Soft bullets may upset enough to give some accuracy.
Going back to the Contender with the .308 barrel, they are usually found with a 1 in 10 twist and that opens them up to using heavier than normal bullets.
In dealing with 32-20s it really depends on which you have; revolver, or rifle, older or newer.