Kāya is by default rūpa-kāya physical body in canonical Abhidhamma Sati-'paṭṭhāna

 

Te Ab Vb 7 Sati-'paṭṭhāna-vibhaṅga

• Te Ab Vb 7 Sati-'paṭṭhāna-vibhaṅga
  • The Analysis of the Ways of Attending to Mindfulness{7}
• 1. Suttanta-bhājanīyaṁ, The Section Derived from the Discourses
  • 01: Kāy-ānupassanā-niddeso Explanation of the Contemplation of the Body
    • gloss: ‘Anupassī.’, ‘Contemplating.’
    • gloss: ‘Viharatī.’, dwells
    • gloss: ‘Ātāpī.’, ardent
    • gloss: ‘Sampajāno.’, lucid-discerning
    • gloss: ‘Satimā.’, rememberful [of Dharma]
    • gloss: loke, world
    • gloss: ‘abhijjhā’, avarice
    • gloss: ‘domanassan.’, distress
  • 02: Vedan-ānupassanā-niddeso, Explanation of the Contemplation of Feelings
  • 03: Citt-ānupassanā-niddeso, Explanation of the Contemplation of the Mind
  • 04: Dhamm-ānupassanā-niddeso, Explanation of the Contemplation of (the Nature of) Things
• 2. Abhidhammabhājanīyaṁ, The Section Derived from the Abstract Teaching
  • [01: Paṭhamanayo]
  • [02: Dutiyanayo]

excerpts:

1. Suttanta-bhājanīyaṁ, The Section Derived from the Discourses

kaya is 31 anatomical body parts, adjhatta = personal body, bahiddha = bodies of others 

01: Kāyānupassanāniddeso	Explanation of the Contemplation of the Body
[356]	[356]
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?
Idha bhikkhu ajjhattaṁ kāyaṁ -	Here a monk in regard to himself -
uddhaṁ pādatalā, adho kesamatthakā, tacapariyantaṁ,	from the sole of the feet upwards, from the hair of the head down, bounded by the skin,
pūraṁ nānappakārassa asucino - paccavekkhati:	and filled with manifold impurities - reflects (thus):{12}
Atthi imasmiṁ kāye:	There are in this body:
kesā, lomā, nakhā, dantā, taco,	hairs of the head, body hairs, nails, teeth, skin,
maṁsaṁ, nahāru, aṭṭhi, aṭṭhimiñjā, vakkaṁ,	flesh, sinews, bones, bone-marrow, kidneys,
hadayaṁ, yakanaṁ, kilomakaṁ, pihakaṁ, papphāsaṁ,	heart, liver, pleura, spleen, lungs,
antaṁ, antaguṇaṁ, udariyaṁ, karīsaṁ,	intestines, mesentery, undigested food, excrement,
pittaṁ, semhaṁ, pubbo, lohitaṁ, sedo, medo,	bile, phlegm, pus, blood, sweat, fat,
assu, vasā, kheḷo, siṅghāṇikā, lasikā, muttan-ti.	tears, grease, spit, mucus, synovial fluid, urine.
So taṁ nimittaṁ āsevati bhāveti bahulīkaroti svāvatthitaṁ vavatthapeti,	He practices, develops, makes much of that sign, and fixes its definition,{13}

gloss: loke, world, note that kaya is part of the world, and rupa (4 elements of anatomical body) is the first of 5 khandhas

[362]	[362]
‘Vineyya loke abhijjhādomanassan.’-ti	‘After removing avarice and sorrow regarding the world.’
Tattha, katamo loko?	Herein, what is the world?
Sveva kāyo loko, pañca pi upādānakkhandhā loko:	For sure it is the world of his own body, the world of the five constituents (of mind and body) that provide fuel for attachment:
ayaṁ vuccati ‘loko.’	this is called ‘world.’

2. Abhidhammabhājanīyaṁ, The Section Derived from the Abstract Teaching

2. Abhidhammabhājanīyaṁ	The Section Derived from the Abstract Teaching

Now we're under the "abhidhamma analysis" section. Note they don't redefine 'kaya' here as mental body, and all four satipatthana are done concurrently with supramundane first jhana!

[01: Paṭhamanayo]

[01: Paṭhamanayo]	[The First Method]{33}
[374]	[374]
Cattāro Satipaṭṭhānā:	Four Ways of Attending to Mindfulness:
Idha bhikkhu kāye kāyānupassī viharati,	Here a monk dwells contemplating (the nature of) the body in the body,
vedanāsu vedanānupassī viharati,	he dwells contemplating (the nature of) feelings in feelings,
citte cittānupassī viharati,	he dwells contemplating (the nature of) the mind in the mind,
dhammesu dhammānupassī viharati.	he dwells contemplating (the nature of) things in (various) things.
[375]	[375]
Kathañ-ca bhikkhu kāye kāyānupassī viharati?	And how does a monk dwell contemplating (the nature of) the body in the body?
Idha bhikkhu yasmiṁ samaye lokuttaraṁ jhānaṁ bhāveti,	Here a monk, at whatever time, develops the supermundane absorption,{34}
niyyānikaṁ apacayagāmiṁ,	which leads out,{35} which goes to decrease (of rebirth),
diṭṭhigatānaṁ pahānāya, paṭhamāya bhūmiyā pattiyā,	to abandonment of wrong views, to the attainment of the first ground,{36}
vivicceva kāmehi, vivicca akusalehi dhammehi,	(where,) quite secluded from sense desires, secluded from unwholesome things,
savitakkaṁ, savicāraṁ, vivekajaṁ pītisukhaṁ,	having thinking, reflection, and the happiness and rapture born of seclusion,
paṭhamaṁ jhānaṁ upasampajja viharati,	he dwells, having attained the first absorption,{37}
dukkhapaṭipadaṁ dandhābhiññaṁ kāye kāyānupassī,	with difficulty in practice{38} and slow deepening of knowledge, and he contemplates (the nature of) the body in the body,{39}
yā tasmiṁ samaye sati anussati Sammāsati Satisambojjhaṅgo,	and that which at that time is mindfulness, recollection,{40} Right Mindfulness, the Mindfulness factor of Complete Awakening,
Maggaṅgaṁ Maggapariyāpannaṁ:	the Path factor, and what belongs to the Path:
idaṁ vuccati ‘satipaṭṭhānaṁ.’	this is said to be ‘attending to mindfulness.’
Avasesā dhammā satipaṭṭhānasampayuttā.	(Also) the rest of the things associated with the ways of attending to mindfulness.{41}
[376]	[376]

...

supramundane first jhana is not a mental frozen stupor, and has awareness of the physical body 

Sammādiṭṭhi hoti, hirī hoti, ottappaṁ hoti,	there is Right View, there is conscience, there is shame,
kāyapassaddhi hoti, cittapassaddhi hoti,	there is bodily calm, there is mental calm, (body and mind contrasted here!)
kāyalahutā hoti, cittalahutā hoti,	there is bodily lightness, there is mental lightness,
kāyamudutā hoti, cittamudutā hoti,	there is bodily plasticity, there is mental plasticity,
kāyakammaññatā hoti, cittakammaññatā hoti,	there is bodily workableness, there is mental workableness,
kāyapāguññatā hoti, cittapāguññatā hoti,	there is bodily proficiency, there is mental proficiency,
kāyujukatā hoti, cittujukatā hoti,	there is bodily uprightness, there is mental uprightness,
sati hoti, sampajaññaṁ hoti, samatho hoti, vipassanā hoti,	there is mindfulness, there is full awareness (as opposed to Vism. frozen stupor!), there is calm, there is insight,
paggāho hoti, avikkhepo hoti:	there is support, there is balance:
ime dhammā kusalā.	these are wholesome things.
Tasseva lokuttarassa kusalassa jhānassa katattā bhāvitattā vipākaṁ,	For one who has produced and developed this supermundane, wholesome absorption, the result (is that),
vivicceva kāmehi, vivicca akusalehi dhammehi,	(where,) quite secluded from sense desires, secluded from unwholesome things,
savitakkaṁ, savicāraṁ, vivekajaṁ pītisukhaṁ,	having thinking, reflection, and the happiness and rapture born of seclusion,
paṭhamaṁ jhānaṁ upasampajja viharati,	he dwells, having attained the first absorption,