Dmytro wrote:"'From birth as a requisite condition come aging and death.' Thus it has been said. And this is the way to understand how from birth as a requisite condition come aging and death. If there were no birth at all, in any way, of anything anywhere — i.e., of devas in the state of devas, of celestials in the state of celestials, of spirits in the state of spirits, of demons in the state of demons, of human beings in the human state, of quadrupeds in the state of quadrupeds, of birds in the state of birds, of snakes in the state of snakes, or of any being in its own state — in the utter absence of birth, from the cessation of birth, would aging and death be discerned?"
"Thus this is a cause, this is a reason, this is an origination, this is a requisite condition for aging and death, i.e., birth.
That's fine. Note though the use of the term "discerned" in the translation, rather than "occur" or "come to be". Yet, how can "death" be known and discerned? As Bhikkhu Nanananda states (p26, Magic Of The Mind), "The law of Dependent Arising in a Noble Norm (ariyo nayo) which in all its twelve-linked completeness is well-seen and well-penetrated through wisdom (pannaya suditto hoti suppatividdho) even by a Stream-winner (sotapanna) who may not possess the knowledge of past lives"
"'From consciousness as a requisite condition comes name-and-form.' Thus it has been said. And this is the way to understand how from consciousness as a requisite condition comes name-and-form. If consciousness were not to descend into the mother's womb, would name-and-form take shape in the womb?"
"If, after descending into the womb, consciousness were to depart, would name-and-form be produced for this world?"
"If the consciousness of the young boy or girl were to be cut off, would name-and-form ripen, grow, and reach maturity?"
"Thus this is a cause, this is a reason, this is an origination, this is a requisite condition for name-and-form, i.e., consciousness."
Again, note that it speaks not only of the first moment of consciousness in this life, but also later in life at the time of the "young boy or girl", firstly indicating that it's not a "once in a lifetime" event indicating conception... rather, that is a repeated process throughout life (as are all the other links). Furthermore, "name and form" should be understood as such... (from the Sammaditthi Sutta of the MN...)
"Feeling, perception, intention, contact, attention - these O friends are called 'name'. The four great element and for dependent on them - these, O friends, are called 'form".
Thus, I do not accept that this step is explicitly talking about a transition from one life to the second, as the "three lives" translation would infer.
Dmytro wrote:22. "And what is aging and death, what is the origin of aging and death, what is the cessation of aging and death, what is the way leading to the cessation of aging and death? The aging of beings in the various orders of beings, their old age, brokenness of teeth, grayness of hair, wrinkling of skin, decline of life, weakness of faculties — this is called aging. The passing of beings out of the various orders of beings, their passing away, dissolution, disappearance, dying, completion of time, dissolution of the aggregates, laying down of the body — this is called death. So this aging and this death are what is called aging and death. With the arising of birth there is the arising of aging and death. With the cessation of birth there is the cessation of aging and death. The way leading to the cessation of aging and death is just this Noble Eightfold Path; that is, right view... right concentration.
I would argue that the "aging and death" described here is that arising from attavadupadana (attachment to the I concept), which leads to sensuous becoming and the birth of the falsely-perceived self. This is of course the strongest, most significant and most powerful "birth" conditioned by ignorance that we will experience in this lifetime... that of me, myself and I. The process, going from ignorance through to death is the "house builder" that the Buddha refers to in this emphatic Dhammapada quote (vv 153-154)...
"Through many a samsaric birth have I ran
Seeking the house-builder in vain
Pain it is to be born again and again.
O! Housebuilder, though art seen
Thou shalt build no house again
Shattered lie all thy rafters
Thy roof-top is torn asunder
Mind attained fermentless state
Reached is craving's end
In other words, having overcome ignorance, the dependent origination sequence has ceases, thus craving has ended, and becoming and birth in their multifarious forms have ceased.
In conjunction with this, consider also the following, again from Bhikkhu Nanananda (p83) ...
"The Buddha realized that 'birth' and 'death' are inseparable collaries of the conceit of existence. The law of impermanence which holds sway even in heavenly realms would militate against any notion of immortal existence. Besides, the quest for immortal existence was only a symptom of the deep-seated fear of death. If only this obsessional fear could be removed the problem would be no more. Hence he advanced a novel type of solution to the problem of life and death. He pointed out that although immortal existence is impossible, one could still experience 'ambrosial'  deathlessness - and that even here and now. One had to recognise fully the truths of impermanence, suffering and not-self whereby 'existence', on which both 'birth' and 'death' depend is made to cease..... Thus instead of attempting to 'stifle' death artificially by heavenly ambrosia, the Buddha saw to it that death died a natural death in a sphere of transcendental experience of a Deathless attainable in this very mortal word"
Existence is to be understood as per SN12.15 at the end of this post.
 Itiv 62 - "Having touched with the body the Deathless Element which is Asset-less and realized the relinquishment of all assets, the Fully Enlightened One who is influx-free, teaches the Sorrowless, Taintless State
No ignorance = no birth, aging and death.
26. "And what is birth, what is the origin of birth, what is the cessation of birth, what is the way leading to the cessation of birth? The birth of beings into the various orders of beings, their coming to birth, precipitation [in a womb], generation, manifestation of the aggregates, obtaining the bases for contact — this is called birth. With the arising of being there is the arising of birth. With the cessation of being there is the cessation of birth
. The way leading to the cessation of birth is just this Noble Eightfold Path; that is, right view... right concentration.
http://www.accesstoinsight.org/tipitaka ... html#aging
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Firstly, I would suggest we take out the crassly inserted "in a womb" that does not actually appear in the text. Secondly, I would suggest we look upon the bolded and particularly underlined "being" section in the context of SN 12.15...
Dwelling at Savatthi... Then Ven. Kaccayana Gotta approached the Blessed One and, on arrival, having bowed down, sat to one side. As he was sitting there he said to the Blessed One: "Lord, 'Right view, right view,' it is said. To what extent is there right view?"
"By & large, Kaccayana, this world is supported by (takes as its object) a polarity, that of existence & non-existence. But when one sees the origination of the world as it actually is with right discernment, 'non-existence' with reference to the world does not occur to one. When one sees the cessation of the world as it actually is with right discernment, 'existence' with reference to the world does not occur to one.
"By & large, Kaccayana, this world is in bondage to attachments, clingings (sustenances), & biases. But one such as this does not get involved with or cling to these attachments, clingings, fixations of awareness, biases, or obsessions; nor is he resolved on 'my self.' He has no uncertainty or doubt that just stress, when arising, is arising; stress, when passing away, is passing away. In this, his knowledge is independent of others. It's to this extent, Kaccayana, that there is right view.
"'Everything exists': That is one extreme. 'Everything doesn't exist': That is a second extreme. Avoiding these two extremes, the Tathagata teaches the Dhamma via the middle: From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.
"Now from the remainderless fading & cessation of that very ignorance comes the cessation of fabrications. From the cessation of fabrications comes the cessation of consciousness. From the cessation of consciousness comes the cessation of name-&-form. From the cessation of name-&-form comes the cessation of the six sense media. From the cessation of the six sense media comes the cessation of contact. From the cessation of contact comes the cessation of feeling. From the cessation of feeling comes the cessation of craving. From the cessation of craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging & death, sorrow, lamentation, pain, distress, & despair all cease. Such is the cessation of this entire mass of stress & suffering."
I want to thank you for your questions Dmytro... that have presented a good challenge. As the Buddha said in DN 15...
Deep is this dependent co-arising, and deep its appearance. It's because of not understanding and not penetrating this Dhamma that this generation is like a tangled skein, a knotted ball of string, like matted rushes and reeds, and does not go beyond transmigration, beyond the planes of deprivation, woe, and bad destinations.
Thus, we should all continue to investigate further until we develop penetrating wisdom and attain the deathless.
I'd like to think that you might also consider what I have presented too, and maybe even take up the challenge I issued to Tilt so that we can investigate further.
P.S. Read also the first quote you provided in the context of the above responses.