what is the meaning of the cryptic formula in the 4sp satipatthana formula? (for the Chinese parallel)

Householder, Upasaka Frank, Interested,

even traditional translations often use mentioned translations, but those can be not assumed of being made by path practicing Venerables.

As far as traced, my person has only seen Ven. Thanissaro transporting it well in/with "in and of itself", also explaining the approach well, meaning not putting in relation to anything. Neither a perception of it, or when rupa or nama "really" is traced => no I, not I, my, not-my-making of what one likes to be mindfull, inwardly, outwardly... no "sakka", giving into, "kaya" group.

May Nyom simply try to known by himself.

One having not attained Jhana or still firm in Sakaya (giving into group, relation - viewing) ditthi, is not capable to understand of what "in and of itself", "considered by itself" ("an und für sich", a sallop used way of speech in German to cut off what is not essence) means.

My person discussed that also with Ven. Ariyavaṃsa Thera on PP, lets see if possible to add a link... here: Satipaṭṭhāna – setting up mindfulness (one may be careful in regard of the danger of sakkaya-ing in philosphy there). So only in and of itself to be investigated. Althought Venerable prefers the traditional translation, discussion and reaction proofs the right understanding and may serve also those used to traditional sanna, sign, word, in regard of the to-liberation-leading essence, meaning, attha, of this grammatical conjunction.

(Note that this Gift of Dhamma is not meant for trade, exchange, stacks, entertainment and akusala deeds, but as a share of merits and continue such for release)

Without answering the question, You also have the tibetan suttas, but ofc they may not be parallel to the suttas in pali. Since those people use sanskrit, you have to search for smṛti or mindfulness directly for instance http://translator-tools.84000.co/glossary-items.html?term=mindfulness

in the The Transcendent Perfection of Wisdom in Ten Thousand Lines, they claim

Application of mindfulness which, with regard to feelings, observes feelings

http://read.84000.co/translation/UT22084-031-002.html#UT22084-031-002-4055

Perhaps there is also some parallel suttas in gandhari? for instance in http://www.buddhism.org/?p=926 http://www.buddhism.org/?p=930 and the books A Gāndhārī Version of the Rhinoceros Sūtra and https://books.google.com/books?id=UoV5Dz20oo4C&pg=PA240&lpg=PA240&dq=mindfulness#v=onepage&q=mindfulness&f=false

The texts in gandhari are here https://www.gandhari.org/a_manuscripts.php

Prajnaparamita in 10,000 lines, that user12901 linked, says the following:

Moreover, Śāradvatī­putra, the maturity of a great bodhisattva being also entails the four applications of mindfulness which are correctly retained by skill in means. If you ask what these four are, they may be described as follows:

(1) Great bodhisattva beings who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to the inner body, continue to observe the physical body, without generating any apperceptions dependent on the physical body. Similarly, those who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to the outer body, continue to observe the physical body, without generating any apperceptions dependent on the physical body. Similarly, those who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to the inner and outer body combined, continue to observe the physical body, without generating any apperceptions dependent on the physical body.

(2) Those who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to inner feelings, outer feelings, and combined inner and outer feelings, continue to observe feelings, without generating any apperceptions dependent on feelings.

(3) Those who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to the inner mind, the outer mind, and the combined inner and outer mind, continue to observe the mind, without generating any apperceptions dependent on the mind.

(4) Those who are diligent, alert, and mindful, after eliminating worldly covetousness and sadness without apprehending anything, with regard to inner phenomena, outer phenomena, and combined inner and outer phenomena, continue to observe phenomena, without generating any apperceptions dependent on phenomena.

The comment says:

The expression “without apprehending anything” [means] without perceiving anything as inherently existing.

Also, the word "apperceptions" here translates the Sanskrit word "vijnapti" which means "interpretation".

As was pointed out in other answers, Thanissaro Bhikkhu translates this piece as "body (etc) in and of itself". In his awesome book, Wings of Awakening, he explains that it means that the meditator must enter "phenomenological mode", where all phenomena are studied as they appear to the raw observation, without adding the typical worldly interpretations and attitudes.

Buddha himself teaches the following practice in Bahiya Sutta:

Bāhiya, you should train yourself thus: In reference to the seen, there will be only the seen. In reference to the heard, only the heard. In reference to the sensed, only the sensed. In reference to the cognized, only the cognized. That is how you should train yourself. When for you there will be only the seen in reference to the seen, only the heard in reference to the heard, only the sensed in reference to the sensed, only the cognized in reference to the cognized, then, Bāhiya, there is no you in connection with that. When there is no you in connection with that, there is no you there. When there is no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress.

The way I understand all this is, Sathipattana is not some separate meditation practice, but actually training to dwell in suchness. Which means, laying aside all preconceptions, biases, attachments, judgements and side-taking. What's left is completely purified perception of things as they are, yathabhuta, without "asava"s.

"seeing body as body" and "body in body" both means "seeing the body according to reality as it actually is". This is to see the body without any additional interpretations or view. Essentially seeing what is there as it is.

This does not mean "you contemplate body in isolation of the 3 other frames of 4sp". You can choose to if you want to as your main focus, but any of the 4sp one cannot do purely in an isolated manner as one influences the other. E.g. if you are doing kayanupassana this influence vedena. Following highlights how the 4 elements (part of kayanupassana) influence feelings.

vedana can certainly be connected with three of the four mahabhuta (elements), which are collectively known as rupa. These three are pathavi (earth element), tejo (fire element) and vayo (air element). Apo (water element) is too subtle to touch, or is so subtle that one finds difficulty in feeling it. Now, to illustrate the connection between vedana and the three bhuta, when you have a cramp, which is the tightening of the muscles, it is pathavi; when you feel temperature of the body it is tejo; when you suffer from pain in the stomach, it may be something to do with vayo, for when the air in your stomach is stuck and finds no vent to get out, you have that experience of unpleasant feeling called pain. The connection of vedana with these mahabhuta makes it all the more manifest and helps one realize its true nature.

Source: Vipassana and Vedana as Understood by a Novice by U Tin Lwin