'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

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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.