"Iaijutsu Duel (to the death): When fought to
the death, the duel objective is to kill the opponent
or inflict the Dying condition on them."
So, either counts - this is a mechanical thing. If you manage to land a 16+ crit, that character dies instantly, and never enters the dying condition, which would create a loophole. But that crit is also quite hard to land, so the Dying condition also counts - it's a mortal injury they would die from without assistance. You might run into a judge who would contest it, but technically per the rules you hit an objective and the duel is over. And since you've lost a duel to the death, if you don't die you'll be ordered to commit seppuku anyway, so the "normal" thought process is why would you stop yourself from dying. (Seriously, there's a common quote from the Hagakure about this)
If you want to throw honor to the wind and possibly screw over several people in your house/family/clan/party, you might try and stop yourself from dying or keep fighting to get revenge on your enemy, because that is also cool and on theme, but does not necessarily change the "outcome" of the duel, since the two dying condition crits are "Agonizing Death" and "Swift Death".