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The Japanese sentence from the Nichureki is on the 'talk' page of the English Wikipedia. Now I don't speak any Japanese at all, but I can submit that sentence to Google translate, and also all individual kanji, to identify what is said where. And I noticed that different kanji are used to describe the move of the Reverse Chariot and Side Mover on the one hand, and the Flying Dragon on the other. This is what made me doubt the moves of the latter are normal unlimited-range slides.

We've tried to update and clear out the very complex history of shogi and its large variants in a Word of Chess (McFarland, 2017). We were (strongly) helped by Erwann Le Pelleter, speaker of Japanese, who had access to several historical sources in this language.

Dai shogi is first mentioned in the Taiki, a diary made between 1135-1155. The first details on rules are given in the Nichureki, a text compiled in 1210-1221 but containing some parts that might be older. The Nichureki contains a description of a small shogi (that we call Heian Shogi) on 8x8 or 9x8 or 9x9, and THIS Dai Shogi on 13x13 that we call Heian Dai Shogi. The description of the moves of the pieces is short. For the Flying Dragon (Hiryu) it simply says "it flies in the four diagonal", if the translation is correct. Indeed, I agree that it is not enough to affirm that this move was the move of the modern Bishop. As HG remarks, it could also be the move of the piece with the same name at Dai Shogi (on 15x15). But it might be something different too as the indications for several other pieces, Copper General, Iron General, Side Mover, seem to describes different moves than the ones later known for Dai Shogi.

The Futsu shodoshu, a prayer anthology published around 1300 also evokes a Sho Shogi (sho=small) and a Dai Shogi, with a Flying Chariot (Hisha) for the 1st time in all shogi history. This Hisha was maybe playing like the Rook. Contrary to what many authors have said by copying each other, this text had no indication on the board size nor the total number of pieces.

The 15x15 form (we call it Dai Shogi) would have been presented in the Shogi Shushu no Zu, a book from 1443 now lost but which is known from copied passages in later historical books, the earliest being the Shogi Zu from 1591. Dai dai shogi, Maka dai dai shogi and Tai shogi were also in the Shogi Zu, and maybe, they were also known in 1443.

Chu shogi, on 12x12 is first mentioned in the mid 14th c. and appears several times in the 15th c.

The Flying Dragon move that also appears in Dai makes sense. It would be odd for the Bishop to appear in Heian Dai without the Rook as well. In fact, I can't recall any other game that has that property.

In Ten Shogi Variants by George Hodges, he says that Maruo Manabe made up the moves for the pieces and that influenced Steve Evans to put the moves in the image above:

The late Maruo Manabe, of Chigasaki, Japan, who during the 1970s and 1980s was widely considered to be the foremost expert on the Shogi variants, studied this problem of the possible moves and promotions of the pieces. Giving credence to later texts and theories, he suggested moves for those pieces not met with in normal Shogi for those who might wish to try out the game. He assumed that all pieces promote to a Gold General at the third rank, except for the Flying Dragon, which adds the power to go one step in the four orthogonal directions. (Bishop becomes a Dragon King)

You're more informed than I am about the dates of things. Wish it were possible to ask the designers of the games about their thought processes and their choices. Oh well.

Also, Tomoyuki Takami posted his thoughts on the game in 2015 (in Japanese) and believes that it was made before Dai, though with the pieces closer together. He's done extensive research on other variants like Maka Dai Dai as well.

I think the historic evidence rules that out; Dai and Maka Dai Dai Shogi were of much later date. The shrunken version of Dai is Chu Shogi.

BTW, it is quite unlikely that the Flying Dragon moved as a Bishop. The only known rule description uses another word for its move as it uses for sliding in the other pieces. Much more likely it meant that it moved as a Flying Dragon from Dai Shogi. As its name suggests... IIRC its promotion is completely made up.

This is just speculation. I wonder if this game was derived from Dai Shogi as a simpler form, itself perhaps a simpler form of Maka Dai Dai Shogi?

The Go Between is placed oddly. It has no effect on the Flying Dragons (aka Bishops), which themselves point at each other.

The piece density is 40%; not improbable (Wa Shogi is 45%) but seems strange.

If there were three less ranks (13×13 board becomes 13×10), the Go Between would become relevant; the Flying Dragons wouldn't attack each other after pawn moves; and the piece density would increase to 52%, making for a tighter game.

There would also be a more natural four rows between the pawns, just as in Dai.

Sadly, there is no way of knowing for sure as the rules of the game are incomplete, though I would be happy to know more. 13×13 may be the correct size if it were played on a Go board on the points (e.g. Ko Shogi for 19×19).

It seems that, like Wa Shogi, Heian Dai Shogi could be played very well with drops.

I can't believe that it's the earliest form of Shogi! I always was
thinking that first Shogi is like modern 9x9, but on 8x8 and without
'rook' and 'bishop', because it looks like other forms of chess, mostly
Thai game 'Makrook' (and, maybe, pawn capture from Chinese or Korean
chess): silver general moves exactly as bishop in Makrook, Shogi lance and
knight are 'normal' knight and rook, but both moves only forward, only
new piece is gold general, who replaced the ferz, and promotion rules looks
like promotion in Makrook (promotion zone is last 3 ranks, and pawns
promotes to piece, wich replaced ferz)!
Or Shogi, wich we know is only Shogi variant, based on chess?
If somebody knows it, please, explain me!

The move of Iron general may be wrong. In Heian Dai Shogi, 'èc«•ssŒãŽO•û' means Iron general also can step square sideways.

Lack of long range pieces on the so large board,  Heian-Dai Shogi is too  boring to play.