Since PCs running DOS and Windows took off in a way that Amiga didn't, the DOS version of Lemmings ended up likely the most widely played, and certainly the one most easily obtained and installed nowadays, and even so stretching back years. Remember that the community had been around (well not all the same people, though some definitely had been here all this time) for over a decade, and back 15 or so years ago if you don't actually have access to an Amiga computer, your success of running the Amiga version decently on whatever emulators were available at the time for PCs was probably not great. (Well, maybe things were different in Great Britain and Amigas probably a little more common; I lived in USA.) I'm certainly not the first person in this community and certainly not the first person to do challenges, but even at the time I started, when Lemmix hadn't even existed yet and there was really just maybe one or two challenge threads discussing free-formed challenges, challenge results were already primarily reported for DOS versions, because that was the version most people can actually play at the time.
As for development of Lemmix, I happened to have the most technical knowledge and tools available to reverse engineer the DOS game. Remember this was more than a decade ago (in fact like 15 or so years). At the time, I'm not sure I can even easily run the Amiga version given the state of emulation back at the time. And I think Erik himself was most familiar with DOS Lemmings. So for all those reasons, Lemmix also became based on DOS Lemmings, and of course its capabilities make doing challenge solutions easier and also enable better sharing of solutions than screenshots or videos (for the latter, remember that youtube hadn't existed 15 or so years ago). Together with challenge enthusiasts like ClamSpammer, the challenges really started taking off with Lemmix being the primarily program used to discover and share solutions.
It had crossed my mind over the years to consider reverse engineering some of the other versions (primarily Amiga and Mac) and possibly get Lemmix to support them. But it'd be a lot of work with relatively little demand, so hard to drum up motivation. Actually, Amiga emulation had come a long way since, so in addition to being much easier now to run the Amiga version decently on your PC through emulation, you can now probably easily do savestates and even possibly input-recording-based movies on an emulator, which in some ways help fill in some of the tooling capabilities Lemmix offers. Thus making it even less interesting to port Amiga mechanics over to Lemmix.
The Amiga, Atari and DOS versions are co-developed around the same time. But yes, early prototyping of the game was done on the Amiga, as was the level editor they had in-house which as far as we know from Mike Dailly, was only ever an Amiga program.
People could start new challenge threads based on other versions like Amiga and keep another set of records, although I'd expect a relatively low level of participation for obvious reasons.