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Yes, I misinterpreted the pseudo-archbishop given here. It could be a compound of Bishop+Bion ("Korean Vao"), but the description is still ambiguous. Maybe it is the "Bion" only. Bion is the name that problemists gave to the diagonal piece that must jump once for moving with or without capturing. (pB in Betza's)
Concerning Man, of course I know how to spell Man in English and I even know that Mann is German. I have some instruction :=) . But some prefer to use Mann with the German spelling to distinguish the chess fairy piece from the human man. Mann is the choice made on Wikipedia for example. https://en.wikipedia.org/wiki/Mann_(chess). Dickins used Man. The Oxford Companion to CHess (Hooper & Whyld) used Mann. I would say that there is no consensus.
Since this archbishop is a compound of a Bishop and an Arrow/Vao, perhaps Arrow Bishop. Or maybe Archer?
Also, General and Commoner are also fairly common names for the Man/Guard/Prince. I only like Prince if it has the potential to become royal (like a backup king.) And I don't like "Man" at all, personally. The others are all fine.
Note that the 'Archbishop' here according to the description is not a common Vao. It says it can jump, and not that it must jump, and it nowhere mentions that it is divergent. I interpret that as the compound of a normal Bishop and a 'Korean Vao'.
Man is spelled with a single n in English.
Like other comments, I would recommend changing the name of non-standard pieces to names which more usual in chess variants and also because the names which are used now are well known to play differently.
Archbishop > Vao. If Vao is not appreciated there is Arrow, Bow. I use Crocodile in all my variants for some criticisable reasons.
Chancellor > Guard or Mann or Prince
Elephant > Camel
Hawk > Dragon King or Admiral or Sailor
Archbishop is widely used for Bishop+Knight, Elephant for Ferz+Alfil, Chancellor for Rook+Knight, Hawk for ADGH.
This is just an opinion and a suggestion, the author is free to decide of course.
This article seems unfinished; there is no Rules or Notes section, and it doesn't explicitly mention which of the rules of orthodox Chess apply, what the depth of the promotion zone is on this extra-deep board, and whether the unorthodox pieces are also available as promotion choice.
As to the initial setup: this would seem more convenient if the Chancellors and Elephants were swapped. As it is, the Elephants compete with the Knights for the same squares during development.
This could use a graphical diagram. You may use the Diagram Designer to create one. Concerning your piece names, Archbishop, Chancellor, and Elephant are very common names for different pieces than you described them as, your Hawk is more commonly known as a Dragon King, and your Elephant is more commonly known as a Camel.
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I don't like either using Man. I agree for Prince.