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O seems to study something different here: 'thickness-constituting stones' rather than 'influence stones'.
No, not really, because he actually defines his particular use of atsumi by using the word 'influence' (seiryoku). It's more that his purpose that is different from yours (moyos rather than josekis).
Your question about counting territory: "I wonder how clearly aware they are of a maintained sente condition." I don't realy understand what you are talking about, but if you mean that the boundaries are marked about by assuming that the outside player plays everywhere in sente (except for the usual gote-gote exchanges on the edges), then that is standard. There are, however, other elements, the most important of which is that areas with aji are not counted. The White area above would not be counted for territory for that reason. In the diagram below, for example, Black's corners are not counted (by a pro) as confirmed territory because of the aji at A and B which results from the nearby White stones. Without nearby White stones the shimari in the upper right would be marked up with 11 points but counted as "approximately 10". The lower left in isolation would be marked up with 17 points but counted as "approximately 15".
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Being imprecise about numbers of points etc. can be excused in verbal talk. In writing, it shows a lack of courage for the fear of being criticised for possibly stating an amount of territory that is 1 point off from others' consensus.
You are reverting to the sort of language that has weakened your case immeasurably before. 'Lack of courage' is a silly phrase to use. It is not even accurate. In the examples above (due to O Rissei) he (courageously?) marks up a precise number of points in a diagram. He just counts them as an approximation, and this is in line with countless pros who advise that approximations are all that are needed (and also that even such approximations are not needed if you can use the method of offsets). Precise counts are, of course, used at the endgame stage, but even there an adjustment will always be made for thckness.
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Cho Chikun has not been the only Japanese pro to consider settled / confirmed / current territory
For you to make this remark, I have to assume that you have only ever seen the method once, in a book in English by Cho. I feel sure it's been used in several other English texts, but in Japan it's mainstream, and goes back a long way. I think I first saw it used systematically by Kato Shin, Kubomatsu and Iwamoto in the 1920s with sporadic uses before then. Neither of the terms 'confirmed' or 'current' is ideal in English but in Japanese, incidentally, where there is no tense in the verb, the phrase appears as 'confirm territory'. The phrase can thus be interpreted as e.g. 'territory that can be regarded as settled' or 'will be settled', etc. This flexibility is a characteristic of the language and has been missed, for example, by those who pore over Japanese rules.
I'm not going to hunt through my library for old references I've gven before, I'm afraid, but the O Rissei book is on my desk at the moment so I can give you the ISBN of that: 978-4-8399-5590-8. There's not a lot of text, but in this case do note that the words are crucial and very interesting (e.g. new terminology: 'adding pith' (芯を入れる) to a moyo as being the correct way to create the right rhythm for building the moyo - and no, reduction is not taking the pith).