According to this answer on Reddit, each sort corresponds to an edition between 1 and 3:
The Japanese edition is the second edition. The Japanese and English edition is also likely the second edition. It’s the multilingual edition that’s usually the third edition.
If there’s actually a difference other than that… well, if you look at other Japanese textbooks, the most ‘multilingual’ editions usually contain translations or notes in English, Chinese and Korean. However, if you don’t see that appearing in any of the Genki editions you find, then that’s most likely not the case.