Dear Pāli friends,
Over the years I deeply researched the term 'sati'. Let me summarize my findings.
In the context of Satipatthana, sati is the taking up of the object-support (ārammaṇa):
where the object-support (ārammaṇa) can be of four kinds, corresponding to body, etc.:
Tattha ārammaṇe pakkhanditvā upaṭṭhānaṭṭhena paṭṭhānaṃ, satiyeva paṭṭhānaṃ satipaṭṭhānaṃ. Ārammaṇassa pana kāyādivasena catubbidhattā vuttaṃ ‘‘cattāro satipaṭṭhānā’’ti. Tathā hi kāyavedanācittadhammesu subhasukhaniccaattasaññānaṃ pahānato asubhadukkhāniccānattatāgahaṇato ca nesaṃ kāyānupassanādibhāvo vibhatto.
In the context of Satipatthana, sati is directly related to functions of memory, and is rightly called 'remembrance' (saraṇa):
Tattha, katamā sati?
Herein, what is mindfulness?
Yā sati anussati ...pe...paṭissati sati saraṇatā,
That which is mindfulness, recollection, ...pe...recall, mindfulness, remembrance,
dhāraṇatā apilāpanatā asammussanatā,
bearing (in mind), not losing, not confusing,
sati Satindriyaṁ Satibalaṁ Sammāsati:
mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness:
ayaṁ vuccati ‘sati’.
this is said to be ‘mindfulness’.
Satiyeva satipaṭṭhānaṃ. Atha vā saraṇaṭṭhena sati, upaṭṭhānaṭṭhena paṭṭhānaṃ. Iti sati ca sā paṭṭhānaṃ cātipi satipaṭṭhānaṃ. Idamidhādhippetaṃ.
Yadi evaṃ kasmā ‘‘satipaṭṭhānā’’ti bahuvacanaṃ? Satibahuttā. Ārammaṇabhedena hi bahukā etā satiyo.
So there's no real difference in meaning of sati between the context of Satipatthana and the context or recollections, remembrances (e.g. of Tathagata) - it's just that sati may have take up various object-supports (ārammaṇa). The unified meaning of sati across the contexts is illustrated in Mahaniddesa:
Satoti catūhi kāraṇehi sato – kāye kāyānupassanāsatipaṭṭhānaṃ bhāvento sato, vedanāsu…pe… citte… dhammesu dhammānupassanāsatipaṭṭhānaṃ bhāvento sato.
Aparehipi catūhi kāraṇehi sato – asatiparivajjanāya sato, satikaraṇīyānaṃ dhammānaṃ katattā sato, satiparibandhānaṃ dhammānaṃ hatattā sato, satinimittānaṃ dhammānaṃ asammuṭṭhattā sato.
Aparehipi catūhi kāraṇehi sato – satiyā samannāgatattā sato, satiyā vasitattā sato, satiyā pāguññatāya sato, satiyā apaccorohaṇatāya sato.
Aparehipi catūhi kāraṇehi sato – sattattā sato, santattā sato, samitattā sato, santadhammasamannāgatattā sato. Buddhānussatiyā sato, dhammānussatiyā sato, saṅghānussatiyā sato, sīlānussatiyā sato, cāgānussatiyā sato, devatānussatiyā sato, ānāpānassatiyā sato, maraṇassatiyā sato, kāyagatāsatiyā sato, upasamānussatiyā sato. Yā sati anussati paṭissati sati saraṇatā dhāraṇatā apilāpanatā asammussanatā sati satindriyaṃ satibalaṃ sammāsati satisambojjhaṅgo ekāyanamaggo, ayaṃ vuccati sati. Imāya satiyā upeto hoti samupeto upagato samupagato upapanno samupapanno samannāgato, so vuccati sato.
(from http://www.dhammawhe...=238781#p238781 )
Similarly in Abhidharma-samuccaya:
"What are the objects (ālambana) of the application of mindfulness (smṛtyupasthāna)? They are the body (kāya), feeling (vedanā), mind (citta) and mental qualities and objects (dharma). Or [they are] things pertaining to oneself (ātmāśraya-vastu), things experienced by oneself (ātmavastu) and qualities pertaining to the defilement and purification of oneself (ātmasaṃkleśavyavadānavastu)."
"The cultivation of mindfulness (smṛtibhāvanā) [is accomplished] by counteracting the minor defilement of confusion (forgetfulness) (saṃmosa) with regard to the teaching of the Blessed One (bhagavatah śāsane)."
This traditional definition of sati as "ārammaṇapariggahita" agrees with the "dhāraṇatā" from Dhammasangani 16:
Katamaa tasmi.m samaye sati hoti? Yaa tasmi.m samaye sati anussati pa.tissati sati sara.nataa dhāraṇatā apilaapanataa asammussanataa sati satindriya.m satibala.m sammaasati– aya.m tasmi.m samaye sati hoti.
And with Milinda-pañha 37-38:
eva.m kho, mahaaraaja, upaggaṇhanalakkhaṇā sati.
"Just so, Your Majesty, does sati have the characteristic of taking hold."
This 'taking up' (pariggahita) of one or another of satipaṭṭhanas as an object-support (ārammaṇa) is expounded in Satipatthana sutta, and explained in more detail in Satipatthana-vibhanga:
Kathañ-ca bhikkhu ajjhattaṁ kāye kāyānupassī viharati?
And how does a monk dwell contemplating (the nature of) the body in the body in regard to himself?
So taṁ nimittaṁ āsevati bhāveti bahulīkaroti svāvatthitaṁ vavatthapeti,
He practices, develops, makes much of that sign, and fixes its definition,
so taṁ nimittaṁ āsevitvā bhāvetvā bahulīkaritvā svāvatthitaṁ vavatthapetvā,
and after he has practised, developed, made much of that sign, and fixed its definition,
bahiddhā kāye cittaṁ upasaṁharati.
he focuses his mind on another’s body.
where "nimitta" (representation, often mistranslated as 'sign') is a type of "ārammaṇa" (object-support), and "svāvatthitaṁ vavatthapeti" is close to "pariggahita".
As explained in the Commentary, only one of Satipatthanas is taken up for remembrance (sati):
"apica tasmiṃ janapade catasso parisā pakatiyāva satipaṭṭhānabhāvanānuyogamanuyuttā viharanti, antamaso dāsakammakaraparijanāpi satipaṭṭhānappaṭisaṃyuttameva kathaṃ kathenti. udakatitthasuttakantanaṭṭhānādīsupi niratthakakathā nāma na pavattati. sace kāci itthī “amma tvaṃ kataraṃ satipaṭṭhānabhāvanaṃ manasikarosī”ti pucchitā “na kiñcī”ti vadati. taṃ garahanti “dhiratthu tava jīvitaṃ, jīvamānāpi tvaṃ matasadisā”ti. atha naṃ “mā dāni puna evamakāsī”ti ovaditvā aññataraṃ satipaṭṭhānaṃ uggaṇhāpenti. yā pana “ahaṃ asukaṃ satipaṭṭhānaṃ manasikaromī”ti vadati. tassā “sādhu sādhū”ti sādhukāraṃ datvā “tava jīvitaṃ sujīvitaṃ, tvaṃ nāma manussattaṃ pattā, tavatthāya sammāsambuddho uppanno”tiādīhi pasaṃsanti.
Further, in that territory of the Kuru people, the four classes — bhikkhu, bhikkhuni, upasaka, upasika — generally by nature were earnest in the application of the Arousing of Mindfulness to their daily life. At the very lowest, even servants, usually, spoke with mindfulness. At wells or in spinning halls useless talk was not heard. If some woman asked of another woman, "Mother, which Arousing of Mindfulness do you practice?" and got the reply, "None at all," then that woman who replied so was reproached thus: "Your life is shameful; though you live you are as if dead," and was taught one of the kinds of Mindfulness-arousing. But on being questioned if she said that she was practicing such and such an Arousing of Mindfulness, then she was praised thus: "Well done, well done! Your life is blessed; you are really one who has attained to the human state; for you the Sammasambuddhas have come to be.""
"kāyānupassīti kāyamanupassanasīlo, kāyaṃ vā anupassamāno.
“kāye”ti ca vatvāpi puna “kāyānupassī”ti dutiyaṃ kāyaggahaṇaṃ asammissato vavatthānaghanavinibbhogādidassanatthaṃ katanti veditabbaṃ. tena na kāye vedanānupassī vā, cittadhammānupassī vā, atha kho kāyānupassīyevāti kāyasaṅkhāte vatthusmiṃ kāyānupassanākārasseva dassanena asammissato vavatthānaṃ dassitaṃ hoti.
Kayanupassi = "Contemplating the body." Possessed of the character of body-contemplation, or of observing the body.
Why is the word "body" used twice in the phrase: "Contemplating the body in the body?" For determining the object and isolating it, and for the sifting out thoroughly [vinibbhoga] of the apparently compact [ghana] nature of things like continuity [santati].
Because there is no contemplating of feeling, consciousness nor mental objects in the body, but just the contemplating of the body only, determination through isolation is set forth by the pointing out of the way of contemplating the body only in the property called the body."
Such remembrance of one or another of Satipatthanas as a whole sphere (gocara) is described in the Satipatthana sutta itself:
Atthi kāyoti vā panassa sati paccupaṭṭhitā hoti yāvadeva ñāṇamattāya patissatimattāya.
Or his mindfulness that 'There is a body' is maintained to the extent of knowledge & remembrance.
Atthi vedanāti vā panassa sati paccupaṭṭhitā hoti yāvadeva ñāṇamattāya patissatimattāya.
Or his mindfulness that 'There are feelings' is maintained to the extent of knowledge & remembrance.
Atthi cittanti vā panassa sati paccupaṭṭhitā hoti yāvadeva ñāṇamattāya patissatimattāya.
Or his mindfulness that 'There is a mind' is maintained to the extent of knowledge & remembrance.
Atthi dhammāti vā panassa sati paccupaṭṭhitā hoti yāvadeva ñāṇamattāya patissatimattāya.
Or his mindfulness that 'There are mental qualities' is maintained to the extent of knowledge & remembrance.
with more details in Dvedhavitakka sutta:
Seyyathāpi bhikkhave gimhānaṃ pacchime māse sabbasassesu gāmantasambhatesu gopālako gāvo rakkheyya. Tassa rukkhamūlagatassa vā abbhokāsagatassa vā satikaraṇīyameva hoti: etaṃ gāvoti. Evameva kho bhikkhave satikaraṇīyameva ahosi: ete dhammāti.
"Just as in the last month of the hot season, when all the crops have been gathered into the village, a cowherd would look after his cows: While resting under the shade of a tree or out in the open, he simply keeps himself mindful of 'those cows.' In the same way, I simply kept myself mindful of 'those mental qualities.'"