The .32-40 never had the word "Special" attached to it. IIRC the Marlins in that caliber were marked simply ".32-40".
Also, while the .32 Special looks like the .30 Winchester Center Fire (.30-30), they are separate cartridges and are not interchangeable. The .32 Special came about because many shooters and hunters liked the .32 caliber. Ballistically nearly identical to the .30-30, it also allowed Winchester to use the same drilling and rifling tooling they used for the .32-20 (.32 Winchester Center Fire) but for a more powerful cartridge. As the .30-30 gained wide acceptance, the .32 Special declined, though many old timers swore it was a lot more powerful than the .30-30.