Thanks for answering!
Hm, the Swedish original (or was it in Latin?) is indeed very important for this kind of interpretation...
But actually I think when we take the English version as the best we got, there still IS the ambiguity of "admitted" in the phrase "No other person can be admitted":
1) admitting stands for ending on the square, or:
2) admitting stands for being on the square, so when moving from e3 to e7 you must be admitted first to e4, then to e5, then to e6 and finally to e7 (in one move of course). When there is one square where the piece is not admitted, like when a piece is standing on e4, or because it's not admitted to e5 since it is the throne, you can't move from e3 to e7.

In the second interpretation a move looks more like "walking" and the first more like "jumping".
So to me the phrase still isn't clear at all.

Rule 2 doesn't help either it seems: does it mean 'you can't jump at all' or 'you can jump as much as you want but just not over the heads of any other piece'?

I can't solve the interpretation dillemma but to me the "walking" interpretation makes rule 11 less "ad hoc" and it would make the rules cleaner/more simple, because then it directly follows from rule 2 and the text of fig.1.
On the other hand, the second sentence of rule nr. 10 looks like the summum of an 'unclean' rule, so I admit it's not a decisive argument.
To me the "walking-not-jumping/passing"-interpretation would be strengthened when it would make the game more balanced, although again it would be circumstantial evidence.