Ok, I found it here:
The idea that the Brihadaranyaka Upanisad was composed by Yajñavalkya doesn’t really match with the history and composition of the texts themselves. Patrick Olivelle, in his introduction to his translation of the early upanisads, gives a brief overview of authorship and chronology and so forth.
For one, the BAU is actually a compilation text pulled from earlier texts that existed independently. You can see this in the separate lineage charts for various sections glued together and also in repeated passages with slight variation across different texts compiled into one. And the texts that do mention Yājñavalkya in the BAU are in third person about him in a lineage of received tradition.
It’s like saying the Anguttara Nikāya was composed by the Buddha: it’s a big overstatement.
And the Upanisads, take the BAU here, often include critiques of rituals and ritualistic brahmins, also saying that those who internalize the ritual in esoteric practices get greater reward than those who perform the ritual as is and get reborn. They also have various important scenes where Kshatriyas are attributed new ideas that the ‘brahmins’ never knew about or were ignorant of. Certainly, the texts were composed and passed down by brahmins, and so the specific literary reasoning behind some of these scenes is not entirely clear, but the Upanisads certainly were not mainstream literature everybody learns about.
If certain recensions and regions had a bend towards more internal readings, that’s different from them being initiated in esoteric internal ritual to escape samsāra. This was emerging, and even in the suttas we see cases of some brahmins who were leading more ascetic and internal lives, while others were wealthy householders involved in normal sacrifice. I’m not saying the BAU is all about escaping samsāra — it’s a diverse text — but much of its material is about esoteric theology, metaphysics and soteriology, etc.
I can’t recall a reference at the time being, but the Satapatha Brahmana was expanded on and added to over time, including after the life of the Buddha. And I believe this includes the BAU, the current recensions of which were likely not tacked on until later. If anyone knows more about this please let me learn 
Many scholars (not all) believe the BAU as we have it today was finished before the life of the Buddha, but it would have been a living text (as is clear internally) for some time and made mainstream later.
That’s my understanding of this.
Well, words mean different things to different people. What matters is that we understand the underlying term sati in the right way. So it’s good that you are reflecting on this!
I can assure you that the satisambojjhaṅga is present in all the EBTs, not just in the Pali.
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My intuition is that the more precise meaning of sati is something like “searching for the correction to the error in the sensory motor predictive model which is the CAUSE of suffering”.
What I mean by that is …
The sensory motor brain evolved because it enabled beings to respond to sensory experience with moves in the world that improved the probability of gene survival.
Beings which predict sensory experience well, survive to pass on their genes.
Beings which predict sensory experience badly, do not survive to pass on their genes.
The sensory motor brain continued to evolve to construct more elaborate sensory motor predictive models that better predicted sensory experience to further improve the probability of gene survival.
Beings which predict sensory experience well, survive to pass on their genes.
Beings which predict sensory experience badly, do not survive to pass on their genes.
How are these superior models constructed?
And how does sati help us to construct them?
When we experience the The 1st arrow of dukkha,
we understand …
This is the arrow of misprediction.
What is the arrow of misprediction?
The arrow misprediction is notification that we have predicted sensory experience incorrectly.
Why have we predicted sensory experience incorrectly?
Because our sensory motor map contains an error.
What name do we give to this error in our sensory motor map?
The name we give to this error in our sensory motor map is āsava.
What is āsava?
Āsava is “saṅkhāra corrupted by avijjā.”
What is “saṅkhāra corrupted by avijjā”?
“Saṅkhāra corrupted by avijjā” is a “sensory motor prediction/hypothesis” corrupted by “not pure knowledge (owing to denial of things as they actually are)”.
What is a “sensory motor prediction/hypothesis”?
The sensory motor brain predicts sensory experience via sensory motor predictions.
The predictions guide our movements through sensory experience constantly but we never actually become aware of them until we make a misprediction.
Then we feel the 1st arrow.
Then we become aware.
The Buddha called these sensory motor precautions, saṅkhāra.
Every sensory motor prediction consists of 3 parts
1: In this place (nāmarūpa)
2: with this aim
3: make THIS move (or think THIS thought)
Illustrated:
First: Pain is experienced.
Then we become conscious (awaken) and seek to orient ourselves in sensory experience by answering the question:
“Where am I?”
Then we respond with:
I am here.
I am at this named form.
Then we seek to further orient ourselves in sensory experience by answering the question: What am I aiming at?
When these 2 things
1: I am here
2: I am aiming at this
“come together”
this is the “saṁ” in “saṅkhāra”.
This is the “together” part of together-move.
saṅkhāra = saṁ + khāra = together-move
When together arises, move arises.
This was a choice when the saṅkhāra was initially constructed.
The choice becomes a “habit” after it is incorporated into the map we use to navigate sensory experience.
The Buddha also referred to this habit as a “home”.
House-holders cling to their homes.
Having oriented ourselves, with saṁ, khāra arises, thus:
Make THIS move (or think THIS thought)
Note: In the brain, (1) physical movements and (2) thinking new thoughts are equivalent because we are just moving from one “place” to another in our map of sensory experience. (see Jeff Hawkins: A Thousand Brains: A New Theory of Intelligence)
These are the 3 parts of a sensory motor prediction AKA saṅkhāra:
1: In THIS place
2: with THIS aim
3: make THIS move
Our map of sensory experience (dhamma) is composed entirely of sensory motor predictions (saṅkhāra)
which either
(1) predict sensory experience correctly (the move fulfills the aim)
or
(2) predict sensory experience incorrectly. (the move does NOT fulfill the aim)
If we correct the errors as they arise, we will experience the 1st arrow but no second arrow.
If we fail to correct the errors and allow them to accumulate, we will experience suffering.
This is ALL suffering.
All suffering, WITHOUT EXCEPTION, is the CHOICE to allow unresolved inner conflicts to accumulate instead of resolving them.
When we notice the 1st arrow in the heart, what is the skillful response?
The skillful response to noticing the first arrow in the heart is to seek to orient oneself in sensory experience by asking the question:
Ask: Where am I?
Answer: “I am here”!
Say: “I am experiencing the 1st arrow.”
Say: “I am experiencing the arrow of misprediction.”
Remove the “I”
The “I” will corrupt the map with self-absorption in the narrative.
The I-infested sensory motor map does not predict sensory experience correctly.
The I-infested sensory motor map leads to suffering.
Anattā!
Renounce the fetter of sakkāya-diṭṭhi!
See the world the way it actually is.
Say: “The first arrow is being experienced.”
Say: “The arrow of misprediction is being experienced.”
Ask: What am I aiming at?
1: To refrain from resisting things as they actually are (and thereby giving rise to the 2nd arrow).
and
2: To, instead, engage sati “searching for the correction to the error in the sensory motor predictive model which is the CAUSE of suffering”.
And how do we engage sati “searching for the correction to the error in the sensory motor predictive model which is the CAUSE of suffering”?
1: First we find the error. (āsava)
2: Then we find a new hypothesis (saṅkhāra) which seems to better predict sensory experience.
3: Then we test that new hypothesis for consistency against all previously observed sensory experience to ensure that no other errors of misprediction arise.
4: If any errors are found, we revise our hypothesis and repeat the process until we find a new saṅkhāra which accurate predicts all known sensory experience.
When we find a new hypothesis which predicts sensory experience more accurately,
we replace the old view
with the new view
in our map of sensory experience
and continue moving forward.
All suffering is ended in this fashion.
Why is that?
Because suffering is the accumulated unresolved errors in our model.
When all such errors are resolved, all suffering is ended.
Why is that?
Because, although the “spoiled brat” part of us petulantly refuses to defer immediate gratification for a desired sensory experience,
the “mature adult” part of us KNOWS that “seeing the world the way it actually is” is more important.
All suffering, without exception, is an inner conflict between the “spoiled brat” and the “mature adult” AKA “learner”.
All suffering is ended by the “mature adult” patiently teaching the “spoiled brat” to restrain the petulant compulsion for immediate gratification of a desired sensory experience in favor of seeing the world the way it actually is so that sensory experience can be navigated with equanimity born of superior understanding.
1: sabbe saṅkhāra annica
2: sabbe saṅkhāra dukkha
3: sabbe dhamma anatta
1: All sensory motor predictions should be treated as temporary BECAUSE
2: All sensory motor predictions are subject to misprediction AND
3: All sensory motor maps are not-self.
Re: not-self: The spoiled brat must be trained not to corrupt the map of sensory experience with self-absorption in the narrative.
The map is not the territory.
By striving to see the world the way it actually is
and not clinging to ANY view,
liberated minds can rapidly construct accurate maps of sensory experience
to solve any problem.
Even the problem of how do we prevent WW3?
IMHO, if the arahants do not return, this will not happen.
We must liberate our minds to stop the world from killing itself.
Liberated minds can solve any problem.
Even the problems of aging, sickness and death.
All problems are solvable with sati. 
In conclusion, sati is the named form we give to the experience which arises when these conditions are present
1: The first arrow has impaled the heart
2: We respond by asking the question: Where am I?
3: We answer the question with “The arrow of misprediction is present”
4: Having thus oriented ourselves in sensory experience, we ask the question: What am I aiming at?
5: We answer the question with: “I am aiming to resolve the inner conflict which gave rise to this arrow.”
We then apply the faculty of “remembering” in THIS context to solve THIS problem.
P.S. A better translation of ajjhattabahiddhā in the “chorus” of MN 10 that makes this MORE CLEAR:
[ajjhattabahiddhā]:
shorthand for: [ajjhattikāni āyatanāni] + [bāhirāni āyatanāni]
(attention directed) towards (an) internal abode + (attention directed) towards (an) external abode
(attention directed) towards sensory motor map vs (attention directed) towards sensory experience
(attention directed) towards the map vs (attention directed) towards the territory
“In this way he abides contemplating in the body the prediction of sensory experience,
or he abides contemplating in the body the observation of sensory experience,
or he abides contemplating in the body the prediction and observation of sensory experience.
Or else he abides contemplating in the body the nature of dhamma arising, [inner conflict discovered]
or he abides contemplating in the body the nature of dhamma vanishing, [inner conflict resolved]
or he abides contemplating in the body the nature of dhamma both arising and vanishing. [inner conflict discovered and resolved]
Or else recollecting that ‘there is a body’ is simply established in him
to the extent necessary for bare knowledge and recollecting.
And he abides independent,
not clinging to anything in the world.
That is how a bhikkhu dwells
contemplating the body as a body.
NOTE: dhamma ALWAYS means sensory motor map AKA sensory motor predictive model.
However, in this context, the Buddha is calling attention to the portion of the map in which the error (āsava) has been discovered.
(dhamma arising) to seize control of the attention and become the focus of “recollecting”
and how this portion vanishes (dhamma vanishing) when the āsava has been destroyed and replaced with a new saṅkhāra which more accurately predicts sensory experience.
Ajahn Sucitto discusses such “dhammas” here:
What about attestations of ‘dhāraṇā’?
I’m not sure precisely what you are asking here.
However, of course, the more stilled the mind, the easier it is to detect the “distill synapse misfire” which we call the “arrow in the heart. Hard to see”.
i.e. The subtlest errors in the sensory motor predictive model calling attention to themselves.