'Kaya-sankhara' in Anapanasati & Paticcasamuppada are not the same.

Question:

The term 'kaya-sankhara' is defined in MN 44 as the 'in & out breathing'.
The same term is found in step 4 of Anapanasati & in the 2nd nidana of Paticcasamuppada, as follows:

He trains himself, 'I will breathe in calming kāyasaṅkhāraṃ. 'He trains himself, 'I will breathe out calming kāyasaṅkhāraṃ.' Anapanasati Sutta

~

And what are saṅkhārā? These three are saṅkhārā: kāyasaṅkhāro, vacī (verbal) saṅkhāro, citta (mind) saṅkhāro. These are called saṅkhārā. Paticca-samuppada-vibhanga Sutta

My questions:

1. Does 'kaya-sankhara' have the same meaning in both contexts, i.e., in & out breathing?
2. If not, is there another alternative definition apart from in MN 44?
3. If not, is it possible for both contexts to have the same meaning?

Answer:

1. No.

Saṅkhāra-paṭiccasamuppāda cause viññāṇā-resultant, nāma&rūpa-resultant, saḷāyatanā-resultant, phassa-resultant, and vedanā-resultant.
Kāya-saṅkhāra is breath, cause of body's living, breath's effect. But breath is an effect, cittaja-rūpa, of paṭiccasamuppāda's elements. Especially for this question, saṅkhāra-paṭiccasamuppāda cause kāya-saṅkhāra, breath, and many other effects.
In cūḷavedallasutta, MN 44, they talking about (1) concentration meditation then (2) three saṅkhāra than nirodhasamāpatti-attainment and (3) nirodhasamāpatti-rising.
Because of above 3 contexts, the 3 saṅkhāra must relate with jhāna. And the proper relating saṅkhāra with saṅkhāra in MN 44 is kāya-saṅkhāra that we can found in ānāpānassati-pabba and ariyavasa-sutta. They are effect-saṅkhāra that are ceased by jhāna.
If breath is saṅkhāra-paticcasamuppāda, arahanta will can't breathe because buddha taught in cessation paṭiccasamuppāda:

With the fading and cessation of ignorance without a remainder determinations cease.

But the fact appear in many sutta that arahanta still breath, except arahanta who achieving 4th jhāna, and nirodhasamāpatti. For the example in dutiya –– ariyavasasuttaṃ:

Bhikkhus, how is the bhikkhu with appeased bodily determinations? Here, bhikkhus, the bhikkhu dispelling pleasantness and unpleasantness and earlier having dispelled pleasure and displeasure, abides in the fourth higher state of the mind. Thus the bhikkhu is with appeased bodily determinations.

2. 1. The context of many sutta related with ānāpānassati, already described itself:

   He trains himself, 'I will breathe in calming bodily fabrication.' He trains himself, 'I will breathe out calming bodily fabrication.'
   Alternative translation: He trains himself, 'I will breathe in decreased breathing effort.' He trains himself, 'I will breathe out decreased breathing effort.'

   Sāriputta described in paṭisambhidāmagga also:
   

   What is "I will breathe in calming bodily fabrication, I will breathe out calming bodily fabrication"?
   What is bodily fabrication?
   Long breathing (that is the first element in ānapāna-pabba) relating with body = That long breathing involve with body, cause of body. The practitioner trains to calm=to stop=to cease that bodily fabrication.

   In abhidhamma, vibhaṅga, is the same.
   For more information, see ānāpānakatha in path of purification.
   
   2. The very important information is: ariyavasasutta that memorized by anuruddha school, ānāpānassatisutta+cūḷavedallasutta that memorized by sāriputta school, and mahāsaṭipaṭṭhānasutta that memorized by ānanda school, relating with each other with out any lacking.
   Why?
   Because they were arahanta who perfectly ceased every bias, then they recited and memorized tipitaka together, not just read.
3. No, they are not the same, they are used in very difference contexts.

Cause-saṅkhāra and effect-saṅkhāra

Tipitaka translation version and reading study system are the main cause of your question. You can't see the answer by yourself because you never recite and memorize suttanta-pāli like the ancient buddhist people did, such as commentary teachers did.

Cause-saṅkhāra and effect-saṅkhāra

There are 2 main type of saṅkhāra words are using in tipitaka: cause-saṅkhāra, that cause effect-saṅkhāra arising, and effect-saṅkhāra, that depending on cause-saṅkhāra to arise.
Cause-saṅkhāra example from S.N. Nidānavagga Paṭiccasamuppādavibhaṅga sutta:

Saṅkhārā paccayā viññāṇaṃ.
Direct translation: Fabrications cause mind.
Alternative translation: Fabrications fabricate mind.

You can see above saṅkhārā is in the cause position of sentence.
Effect-saṅkhāra example from D.N. mahāparinibbānasutta:

handadāni bhikkhave āmantayāmi vo vayadhammā saṅkhārā appamādena sampādetha.
Behold now, bhikkhus, I exhort you: All compounded things[saṅkhārā] are subject to cease. Strive with earnestness.

You can see above saṅkhārā is in the effect position of sentence because buddha taught in S.N. Nidānavagga Paccayasutta:

viññāṇaṃ bhikkhave aniccā saṅkhatā paṭiccasamuppannā khayadhammā vayadhammā virāgadhammā nirodhadhammā. ime vuccanti bhikkhave paṭiccasamuppannā dhammā.
saṅkhārā bhikkhave aniccā saṅkhatā paṭiccasamuppannā khayadhammā vayadhammā virāgadhammā nirodhadhammā. ime vuccanti bhikkhave paṭiccasamuppannā dhammā.
avijjā bhikkhave aniccā saṅkhatā paṭiccasamuppannā khayadhammā vayadhammā virāgadhammā nirodhadhammā. ime vuccanti bhikkhave paṭiccasamuppannā dhammā.
Monks, mind are impermanent, produced by a combination of causes, arise on account of a cause, a wasting thing, a decreasing thing, a fading thing and a ceasing thing.
Monks, fabrications are impermanent, produced by a combination of causes, arise on account of a cause, a wasting thing, a decreasing thing, a fading thing and a ceasing thing.
Monks, ignorance is impermanent, produced by a combination of causes. Arise on account of a cause, a wasting thing, a decreasing thing, a fading thing and a ceasing thing. (Āsavā cause ignorance.)

From above sutta, saṅkhatā (meaning: "effect is fabricated by causes") and khayadhammā is predicate of avijjā, saṅkhārā, and viññānaṃ. Because they are arisen by their causes. So the context of above sutta showing the condition of causes and effects to let the listener see avijjā, saṅkhārā, and viññānaṃ as effect-saṅkhāra like saṅkhārā word in mahāparinibbānasutta, above.
If you understand pāli's relation of saṅkhāra, like I have explained it above to you. You can see the role of karma. Also, you can notice the error of @Dhammadhatu's answer, too.

Role of karma

Role of karma already defined in pāli, S.N. Nidānavagga paṭiccasamuppādavibhaṅgasutta, as cause-saṅkhāra:

Saṅkhārā paccayā viññāṇaṃ.
Direct translation: Fabrications cause mind.
Alternative translation: Fabrications fabricate mind.

You can search more more detail of karma's role by it's nature:

‘‘katame ca, bhikkhave, saṅkhārā? tayome, bhikkhave, saṅkhārā – kāyasaṅkhāro, vacīsaṅkhāro, cittasaṅkhāroฯ ime vuccanti, bhikkhave, saṅkhārāฯ
"And what are fabrications? These three are fabrications: bodily fabrications, verbal fabrications, mental fabrications. These are called fabrications.

You can notice all above saṅkhāras' explanations are causes, paṭiccasamuppādā, of their effects, paṭiccasamuppannā, by bhūmijasutta in the same cannon, S.N. Nidānavagga. This sutta show that "kamma is intention, cetanā, and intention is saṅkhāra", so in many sutta taught about kāya-saṅkhāra (kāya-kamma, kāya-sañcetanā, kāya-ducarita, kāya-sucarita), vacī-saṅkhāra (vacī-kamma, vacī-sañcetanā, vacī-ducarita, vacī-sucarita), citta-saṅkhāra (mano-kamma, mano-sañcetanā, mano-ducarita, mano-sucarita) by the same context as kamma [cause] of vipāka [resultants]. You can use my example pāli words ad search by yourself.
Also, according to that bhūmijasutta, viññāṇa, nama-rupa, saḷāyatana, phassa, and vedanā in the Dependent Origination, are vipāka of saṅkhāra, too. Because no one can say "while viññāṇa arising, there is no nama-rupa arising, no āyatana arising, no phassa arising, and no vedanā arising". So, in M.N. uparipaṇṇāsa, anupada sutta taught the simultaneous arising of them:

[155] Idha bhikkhave sārīputto vivicceva kāmehi vivicca akusalehi dhammehi savitakkaṃ savicāraṃ vivekajaṃ pītisukhaṃ paṭhamaṃ jhānaṃ upasampajja viharati . ye ca paṭhame jhāne dhammā vitakko ca vicāro ca pīti ca sukhañca cittekaggatā
ca phasso vedanā saññā cetanā viññāṇaṃ 1- chando adhimokkho viriyaṃ sati upekkhā manasikāro tyassa dhammā anupadavavatthitā
honti tyassa dhammā viditā uppajjanti viditā upaṭṭhahanti
viditā abbhatthaṃ gacchanti . so evaṃ pajānāti Evaṃ kirame dhammā ahutvā sambhonti hutvā pativentīti 1- . So tesu dhammesu anupāyo anapāyo
anissito appaṭibaddho vippamutto visaṃyutto vimariyādikatena
cetasā viharati . so atthi uttariṃ nissaraṇanti pajānāti tabbahulīkārā atthi tvevassa hoti.
“Again, bhikkhus, Sāriputta, overcoming thoughts and thought processes, the mind internally appeased in one point, with joy and pleasāntness born of concentration abides in the second jhānaThese things of the second jhāna such as internal appeasement, joy, pleasantness, one pointedness of mind, contact, feelings, perceptions, intentions, interest, resolution, effort, mindfulness, equanimity and attention, follow one after the other, to him. They rise, persist and fade with his knowledge. He knows, these things come to be and cause feelings to rise. When these things follow one after the other, he abides with a mind that does not settle, is not bound, is released and unyokedand is unrestricted. knows there is an escape beyond this. With much practise they come to him.

This sutta taught about jhāna, so simultaneous arising in this sutta referring to cause-saṅkhāra. However, pāli-student should use simultaneous arising with effect of the cause-saṅkhāra in the Dependent Origination, too, because when someone birth they must have viññāṇa, nama-rupa, saḷāyatana, phassa, and vedanā. Except asaññasatta-brahmma, arūpa-brahmma, and saññāvedayittanirodhasamāpatti-person, no one live without them, right?

There is only dhammadhatu's user error, no pāli error

But saṅkhāra in M.N. 44 cūlavedallasutta, that @dhammadhatu make error answer in this question-topic is not cause-saṅkhāra, so the pāli-student can not compare saṅkhāra in M.N. 44 cūlavedallasutta with saṅkhāra in bhūmijasutta and paṭiccasamuppādavibhaṅgasutta. So the fact is "it is only @dhammadhatu's error. It is not pāli's error".

Explanation of saṅkhāra in M.N. 44 cūlavedallasutta

M.N. 44 cūlavedallasutta taught about effect-saṅkhāra, because visākhā-upāsaka was anāgāmi-ariya, and dhammadinnā-therī was arahanta-ariya, and both ariya already ceased five kāmaguṇa, so visākhā-upāsaka asked dhammadinnā-therī about jhāna which causes the cessation of five kāmaguṇa. And jhāna causes the cessation of in-&-out breaths, directed thought & evaluation, and perceptions & feelings, too. Therefore dhammadinnā-therī answered:

Assāsapassāsā kho āvuso visākha kāyasaṅkhāro vitakkavicārā vacīsaṅkhāro saññā ca vedanā ca cittasaṅkhāroti.
"In-&-out breaths are bodily fabrications. Directed thought & evaluation are verbal fabrications. Perceptions & feelings are mental fabrications."

Because 4th jhāna causes the cessation of in-&-out breaths, 2nd jhāna causes the cessation of directed thought & evaluation, and saññāvedayitanirodhasamāpatti causes the cessation of perceptions & feelings [saññāvedayitanirodhasamāpatti translation: "the achievement to cease the perceptions & feelings"]. So in D.N. Dasuttarasutta, ten Ariyan methods of living, taught about bodily fabrications (kāya-saṅkhāra) are ceased by 4th jhāna. Also, bodily fabrications (kāya-saṅkhāra) use in ānāpanassati-jhāna method in mahāsatipaṭṭhānasutta, too. Both sutta enough to show that bodily fabrications in M.N. 44 cūlavedallasutta referring to effect-saṅkhāra.
Then, you will not do an reader error like @dhammadhatu done:

AN 6.63 states: "kamma is intention". In the Dependent Origination, intention (cetana) is first mentioned at nama-rupa (4th condition).
It is one of the greatest errors & corruptions to regard "sankhara" (2nd condition) as "kamma" because sankhara (in SN 12.2) is defined (per MN 44) as the in & out breathing (kaya sankhara); initial & discursive thought (vaci sankhara); and perception & feeling (citta sankhara). Bhikkhu Thanissaro explains this very well in the meditative examples in his book The Shape of Suffering.
Suttas, such as SN 14.12, show clearly that intention & kamma do not occur at sankhara (2nd condition), where sankhara is distracting thoughts & perceptions produced by ignorance (where ignorance, the 1st condition, includes the sensuality element or asava), as follows

So, I often say "Tipitaka translation version and reading study system are the main cause of your question. You can't see the answer by yourself because you never recite and memorize suttanta like the ancient buddhist people did, such as commentary teachers did."