The AD&D 2nd Edition Monstrous Manual gives more examples of oozes, from which we can draw some inferences. Oozes, slimes, and jellies are listed together (pg. 276-280.) Puddings have their own separate listing (pg. 297.)
Slimes are largely immobile and plantlike
As in 5th edition, green slime is a dungeon hazard, but here it appears with the more dangerous olive slime. Neither has real movement capabilities besides falling on someone who walks past. The olive slime can form a symbiotic relationship with a host to move around, but the slimes themselves have essentially no agency.
Oozes and jellies are mostly interchangeable
Both are sometimes translucent (crystal ooze, mustard jelly.) They don't seem distinguished by their climbing ability; in 5th Edition both can climb, in 2nd Edition some variants of both can't.
Both reproduce by budding or splitting. Mustard and ochre jellies can split in combat, but stun jellies can't; however, stun jellies appear more closely related to gelatinous cubes. It's unclear whether this is significant.
Puddings are special
Given that puddings have their own entry, they seem to be in a class of their own. 2nd Edition lists four puddings (black, white, dun, and brown) which all modify the same basic description. They seem to have the best of all the generic ooze features, including amorphous movement, splitting, spider climb, and the ability to melt things. Given that everything in that list also exists in other oozes, there doesn't seem to be a mechanical separation between puddings and oozes/jellies.
Instead, puddings seem to just be a special type of oozes that are considered more deadly than the typical ones. They are called "Pudding, Deadly" in their manual section, and have higher XP rewards (similar to their 5th Edition lonely counterpart.)