Mike Okuda specifically stated that they were cargo containers from the Enterprise. They were just filled with the Botany Bay's stuff.
IIRC, Mike Okuda didn't have a thing to do with STII.
Mike Okuda also said that the red Enlisted personnel jumpsuits worn by NCO's in Star Trek II through VI were Cadet uniforms and the people wearing them were cadets. Despite the fact that Yeomen Burke and Samnoe wore them in ST VI. And the fact that cadets wore the same officer jackets that the officers wore except their undershirts were red, instead of the department colours worn by commissioned officers. So I'd take whatever Okuda says about ST II with a grain of salt.
Also, I always thought that the Cargo containers were from the Enterprise, but the Botany Bay was also sent down to the surface but dismantled for supplies for Khan and his followers to build permanent shelters. It's just we never saw what was left of the Botany Bay, either because it was obscured by the sand and dust, or the Enterprise cargo containers were the only thing to survive the devastation caused by the explosion of Ceti Alpha Six.