What’s the difference between a jelly, a pudding, and other oozes?

Several types of named oozes exist in 5th edition, but the naming conventions are strange. Black puddings and ochre jellies exist, alongside simple grey oozes. Do these names signify anything? Obviously the color portion signifies the color of the ooze, but the second halves of the names are more ambiguous.

Since my searches within 5th edition have turned up nothing, previous editions’ lore is welcome.