Which region is the best in terms of lore and design?

In terms of impact on defining and building the world, Kanto would arguably be the most substantial, being the first game in the series and the one that introduced the world. So many of its conventions define Pokémon regions to this day (for better or worse). While it originally may have represented the real world Kanto (or an alternative version of it), with references to real world locations and animals littered throughout the region, character dialogue and, perhaps most prominently, the Pokédex, later games would quickly distance themselves from it and go on to redefine the Pokémon world as its own separate entity.

In this regard, Johto would perhaps be the region to shape the Pokémon world as a distinct world, being the sequel (perhaps intended as the conclusion in some ways) to the Pokémon story. The region name itself immediately differentiated itself from Kanto by not being taken from a real world area, though the geography and culture was still based on the real-world Kinki region. It added a lot more lore to the world, but where Kanto centred around modern developments and futuristic technology (for the time) such as computers and genetic engineering, Johto looked to the past with its history surrounding the legendary [dogs/cats/beasts], Team Rocket's own lament over its lost leader and prominence, the traditional Japanese aesthetic of the region, Celebii's ability to travel through time, and even the series own young history with a return to Kanto and its protagonist.

Hoenn did add lore and legendaries of its own, but to me I think those games were mostly focused on redefining Pokémon and its gameplay as a series in it for the long run, rather than the couple (well, six technically) of stand alone titles that came before. It was a transitionary period in Pokémon, and perhaps its first lapse after the Pokémania phenomenon.

It did make the box legendaries a big part of the story, rather than side quests and optional bosses. It also made them version exclusive, meaning you either had to buy both versions or find someone willing to give up their legendary if you wanted both, where before all exclusives were catchable in multiples and, as of gen 2, breedable.

I have stated many times that I am not a gen 3 fan, and I think a big part of that is that so much of it feels like the world and story are there to facilitate gameplay experiments.

Show off the new GBA graphics and weather effects on routes and with the legendaries. Try out the next stat mechanics for Pokémon and moves. Keep everything more or less the same as before, just a little prettier. Stick in an evil team because Pokémon has to have that, now and forever. Heck, make two.

As an aside, this was probably Pokémon's best chance to move away from the "Team X" formulaic antagonist plotline and show the variety of possible stories that they could tell. Team Rocket was the gen 1/2 plotline, not a mandatory part of each plotline and region.

But no. They literally doubled down. Mandatory evil teams forever. Even if they're not evil. Gotta have a team. Teams are Pokémon. I really hope they can unshackle themselves from that obligation in the near future. Maybe even Winds and Waves. Here's hoping...

Anyway...

Sinnoh is often cited as the region of myths and legends both in-world and outside it. We got (possibly literal) creator gods and their Pokémon avatars... or something. The infamous library stories that implied a deeper connection between humans and Pokémon, and later the one that leaked and saddled Typhlosion with all of those accusations.

Pokémon Legends Arceus gave us the deepest insight into Pokémon's history ever, literally taking place in the past. And though at the end of the day it still ended up being a modern-setting Pokémon game, balls and all despite lore contradictions, it has given us our only real look at the Pokémon world outside of a contemporary setting.

Gen 4 is the generation where all of the core Pokémon mechanics were really nailed down and took their modern forms. In many ways, I think it also established a potential origin point for the world they were building and set the template moving forward.

You know... that and Kanto. Just make gen 1 again and again forever. Choose 1 of 3 starters, get badges, fight the evil team, catch the legendaries, beat your rival, beat the Pokémon league, become the champion, complete the Pokédex...

Unova was also a first, showing that the Pokémon world wasn't just not-Japan. It could use other places, countries and cultures, too. I have argued that these games have the best and most thoughtful stories (for Pokémon), but I feel that by this point the formula had become very predictable.

Introduce a new region based on a real-world region or country, add some legendaries and backstory, and introduce some regionwide, or perhaps worldwide, existential threat to overcome, usually huberistically wielded by the evil team. The threat is overcome and by the next game no one knows or cares that it happened, because it happened in an entirely disconnected region and they have a brand new threat, team and batch of legendaries to push.

So in summation, I would probably say Kanto and Sinnoh did them most to define Pokémon's lore. A long post for an obvious conclusion.