Mystery Indian Variant

Well, the one-time Knight jump for King is also mentioned in the Chaturanga description on chessvariants.com. IIRC they mention the condition that the King should not have been checked before. It is not clear to me (neither in Chaturanga nor in Cambodian) whether this one-time Knight move can be a capture.

If the Bishop and Queen move as in modern Chess, this could be a similarly modernized version of Chaturanga. The presented game does not provide any hard proof that they do, however. One assumes that the one the person telling this story would have complained about it if they didn't, as he would have found out soon enough. (Or that some of the Indian guys commenting on the story would have mentioned it.)

This would not have come back from Persia, though, as Shatranj is very well documented, and the Elephant there definitely only had the two-square jump. Modern Bishops appeared in Europe only with Courier Chess. In Shogi they are much older, though; Dai Shogi is full of Bishops, Queens and similar sliders.

The Bishop definitely moves one square in that game. So if it isn't a modern Bishop it could be a modern Elephant (step one or jump two diagonally). I don't know if this was one of the possible moves in Chaturanga; it could have been their own answer to the awful weakness of the Shatranj Elephant.