Since it seems that most of editions are lost, I'll comment, considering only the 1st, the 2nd and the 4th. All editions have some nice advantages. Despite the borders, restricting king's movement, were added because author felt 8x8 board too small and 10x10 board too big, I feel pity that they were abolished in 8x8. Because they seemed quite fitting this kind of rook: king and pawns, the most important pieces of Quang Trung, have less possibilities to hide from the rook.

I would propose the 8x8 variation that have the limited roads, like in 2nd edition, but here - 4-files-wide, and also using knights at places of queens, and queen at place of knight, for obvious reasons. It have the following benefits:

1) That I said above about hiding from rook.

2) It keeps an ironical moment from past editions: rooks face each other, but can't capture!

3) It gets rid of requirment to have two sets of Xiang-Qi for this. One set of either western or eastern chess will be enough!

4) Why one should have two of seemingly strongest kind of pieces, while having only one of not-so-strong? Or this arrangement have something to do with Vietnamese culture?

Another possible addition is restrictions of queen in same way as king and pawns: it makes her a weapon against these important pieces, yet being vurnelable of other pieces! But maybe, it's an excess feature...

 

What I would like to say about Quang Trung in general - is that I like idea of making a national chess variant for a country that don't have it, and Vu Q Vo fulfiled this idea quite elegantly, making his game(s) really unique, almost "what could be if Vietnam had a CV as different from other variants, as Xiang-Qi, Changi, Shogi, Makruk and western chess are different from each other". It even would be nice, if 100 years later Quang-Trung would be mentoided among others (of course, being the last, as a subclause, with obvious disclaimers). But it's too optimistic, since now Google don't know much about Quang Trung chess... I would like to know, how it were met in Vietnam IRL. Are the any enthusiasts at least?

 

Of course, it's not the only case of making national chess variant. Vu Q Vo made it the most straightforwardly, and with a lot of elegance, but there are others. Some are made by people of that specific nation, some - by people inderectly connected with that nation, and some - by people with no connection with it, just these who like that nation.

In Russia, it were done at least twice, in quite different ways.

The first were the Chess-Battle, not literally Russian, but Soviet - the new type of chess for the new nation, with a strong connection with modern warfare.

The second, opposingly, tryed to follow the spirit of "ancient Russia" - that's Russian Chess, giving archaic names for standart pieces, and using way of capturing from old Russian tower checkers variant (and, perhaps, promotion of king's pawn to Q+N compound, reflects the fact, that queen had this movement for a long time in Russia). Unfortunately, it seems to pretend being real ancient form of chess.

Bulgaria also had a mystification, claiming to be authentic - Boyar Chess, using some really exotic pieces.

Charles Gilman made a Jewish-themed family of variants - Anglojewish Chess. He also have some variants, related to certain locations in Britain. And in general, Gilman himself, with his monstrous classification of pieces, looks like a kind of "Lewis Carroll and JRR Tolkien of chess world".

And there also been quite an elegant variant strongly-themed after Britain itself - Caïssa Britannia by Fergus Duniho.

Australian variant with funny pieces, reflecting it's animals, tribesmen and animals, is Outback Chess by Timothy Newton.

American Chess is, I guess, supposed to be something Americanesque, as the name suggests. It have some unique features, including winning conditions, an alternation for kings.

Hanga Roa by Hernán Marcelo Domínguez Placencia and Juan Pablo Schweitzer Kirsinger also offers something unique, reflectingthe building of Moai on the Easter Island.

Despite India is the likely and widely-belivied motherland of chess, and have several own local variants, she also were given her made-up variant: Ramayana Chess by Luiz Carlos Campos, with islands, mythical characters and leaping-only pieces (which somewhat reflects the spirit of Indian chess).

I'm not going to list all Shatranj with variants with somewhat Middle-Eastern theme.

And I'm personally thinking of Aztec (or Mesoamerican in general) chess, with links to the local culture, mythology, society and non-chess boardgames. However, I have to think about pieces, to make them elegant enough, and not being a mere copy of western chess pieces (which was successfully done by most of variantists I listed above).

(Actually, my favourite "exotic region" is Polynesia, but there's already Hanga Roa chess, plus, how I see a Polynesian chess variant, looks really like Piratecniks, so I better to think about Mesoamerican one, the region, I'm currently interested in.)