Many bands have relied on a member whose role changed significantly between songs or who didn't have a standard instrumental role to help define their sound, a proverbial joker in the deck.
For example, The Rolling Stones had Brian Jones (slide guitar, harp, anything else he could get his hands on). The Velvet Underground had John Cale (classical piano, viola, and bass). Roxy Music had Brian Eno (synthesisers), the 13th Floor Elevators had Tommy Hall (electric jug).
Who of this sort of musician would you say had the most significant role of defining his band's sound? I'd lean toward John Cale. His versatility was crucial in the different moods between songs on the Velvet Underground and Nico. That record had everything from ersatz soul (There She Goes Again) to druggy drones (Waiting for the Man, Heroin, Venus in Furs) to pop ballads (Sunday Morning, Femme Fatale), not to mention All Tommorrow's Parties.
Good topic. Although they had standard instruments they were known for, there was Entwistle on bass and various brass, Macca on acoustic, piano, bass, even Jagger with talking drum on Dance, Worried About You's elec piano, harp and acoustic slide on Back Of My Hand, etc.
I would go with John Paul Jones- the electric piano tremolo sounds defining No Quarter, his bass playing on the blues and rockers, the Mellotron strings/flutes on Kashmir, Rain Song and Stairway, the mandolins(?)/acoustics on their acoustic set stuff, the organ on Since I've Been Loving You.