The official Japanese rules are the 1989 edition, with an English translation available here: Article 1. The game of go.
What you linked to is the commentary for the so-called “2003 Japanese Rules”, which are not officially sanctioned by the Japanese go association, but rather is the sole invention of Robert Jasiek and represents his opinions and views of how the Japanese rules should be redefined in a very technical manner.
I trust that you understand this, but I just wanted to make it clear for anyone else reading along and clicking on these links, since the “2003 rules” are presented with an air of formality that might confuse some people.
The players have the option of resolving these positions through actual play, however, that might not be the strategically optimal way of handling every kind of position. A trivial case would be if player A disputed that their obviously dead stones were alive, then resuming play for player B to resolve the dispute by playing more stones to capture those dead stones would erroneously reduce player B’s score. Instead player B should demonstrate that the stones are dead without resuming play and needlessly filling in their own territory to capture dead stones.
A more interesting case can arise when there is both a bent-four-in-the-corner and an unremovable ko threat. I talk about how this is handled under Japanese rules in this thread: Life and Death under Chinese Rules (see the end of the first post and post 9). In this case, if there were a dispute at the end of the game, resuming play would be the wrong thing to do, since all that black (for the diagrams shown in that thread) could do is start an unwinnable ko fight. However, the proper way to handle this situation is for black to appeal to the special ko rules during the confirmation stage, which allows him to call white’s position in the top-left dead, while also not disturbing the seki in the bottom-right. This is vital for obtaining the best status and score.
This statement seems a bit unclear to me, but I just want to reiterate that the confirmation of living/dead stones depends on the consideration of ideal, hypothetical play with special ko rules, and is not the same as resuming alternating play with the normal ko rules. Hence, confirmation happens in a “stopped game” state. While the players have the option to resume play (per clause 9.3), that might not be the strategically best thing to do. Again, resolving life/death disputes should not require resuming the game (as done on OGS), since the wrong type of ko rules are applied.
It is difficult to speculate exactly why since he does not appear to fully elaborate on that point on that page. However, we should keep in mind that that is just the personal opinion of one person within the context of arguing for abandoning the official Japanese rules text in favor of his own reinterpretation. Note that he also calls clause 13.1 (both players losing) superfluous, but I think the official commentary provides a reasonable explanation for why that rule exists.
I think determining which stones to remove (i.e., the logical confirmation of which stones are alive or dead) is essential before the mechanical act of removing dead stones and counting. I think maybe the OGS system might not do the best to clarify the distinction, but I would interpret 9.2 as saying that the game only actually ends fully after the players have agreed on stone status and gone through the mechanics of removal and counting. Other commentary attached to the rules text supports that view (see for example the discussion about clause 13.1, where it implies that “agreeing to end the game” is a phase separate from making two passes to stop the game.).
It seems that your declaration of illegality hinges on what I believe is an erroneous assumption that two passes equals end of game, rather than just a stoppage, or that the OGS system necessarily forces this false equivalence. Thus, perhaps your view is letting (the possible limitations of) the OGS system dictate the rules rather than the other way around, which would seem more appropriate.