This is mostly wording, not sure [...] there is a definition ...

Words are important, especially when it comes to definition.

The "haircut rule" exists because it is implemented in the code (and the code is the law).
As we are in the computer science scope, the definition of "rule" here is the scientific rule of inference: "a term in logic for a function which takes premises and returns a conclusion", i.e. "if this then that" in layman's language.

In the case of the haircut rule, the "this" is "the debt ratio has exceeded a threshold" and the "that" is "HBD printing rate change". This is what is defined in the code and it makes no reference to the HBD price.

Everything else is a side effect of the application of the rule but is not the rule. So referring to "HBD is no longer one dollar" is not referring to the rule but one of its potential effects.

I have pointed out in [...] several places and [in] the cited sentence that we are talking about the on chain HBD price, not the market price. The price that the blockchain values HBD when converting to HIVE.

The price at which the blockchain converts HBD to HIVE is proposed by the witnesses, which in turn is based on their observation of the price of HBD on the external market. We can therefore deduce that the price of HBD is defined by the market and not by the blockchain itself. So I have trouble conceiving an "on-chain HBD price". But I wouldn't dwell on this subject because in this topic it is indeed more a question of semantics.

That doesn't change the fact direct influence of the haircut rule is on the printing of HBD, not on its price.

For this part I think its important to distinguish the total HBD supply and the available (?) HBD supply...

In this part of your answer, you are going off-topic. Your paragraph concerns "let's take a look at all the ways HBD is printed". The fact is that one was incorrect and one was missing, even if we agree that the DHF Budget does not enter into account in the calculation of the debt ratio.

There in lower and upper soft limit. The upper soft limit is now taken down to the lower soft limit, but the hard limit remains.

You get the point here.
I missed the fact that the "soft-low" and "soft-high" thresholds were set with the same value since HF26 (config.hpp/ L301 for those who want to check the code). Previously the "soft-high" threshold was the same as the "hard" threshold. There is therefore effectively a sudden stop in the printing of HBDs as soon as the debt ratio exceeds the "soft-low" threshold.

... when DHF payouts are made, that increases the available supply and the debt levels.

The title of your section is "But what is happening to the HBD printing?" and you later refer to "DHF payouts" as "Another source for HBD", which is incorrect.

Even if we agree that the DHF payouts have an influence on the "available" HDD supply and the debt ratio, although I can understand your reasoning, I think that you are mixing several concepts here, which only adds confusion.