A Howdah pistol was designed to be fired from a howdah saddle on an elephant. Or a short barrelled rifle or shotgun. Against eg a tiger on the head or tail of the elephant.

IMO short barrelled. Large calibre.Muzzle loading or breech loading. Has hammers. Side by side.

Have no idea where single barrels idea has come from. Or under and over firearms.

Are their actual examples of an under and over handgun used from an elephant from India from early colonial days?

Most of these are coach or gentleman's guns, against banditry, and also usually smaller calibres. Reportedly renaming such an antique as a howdah adds to the auction price. For example the flintlock under and over pistol displayed in the movie "The Ghost and the Darkness" is a gentleman's handgun. Did the gentleman in a howdah plan to use the fold out bayonet to poke the eye of the tiger ?

No doubt some single shots were used from howdahs in the days of yore. Cheaper. And also in the very early days single shots would have been far more common with all sorts of BP muzzle loaders.