'Paṭicca+sam+uppāda' means 'dependent+co+origination'. It refers to 12 conditions that arise together to result in sorrow, lamentation, pain, grief & despair, as explained in the scripture SN 12.2.
It others words, it does not describe some conditions in the past, some conditions in the present & some conditions in the future and it is not about reincarnation or past and future lives. It is about 12 conditions that arise in the present that result in sorrow, lamentation, pain, grief & despair. The Buddha said about dependent co-origination:
Good, monks. You have been guided by me in this Dhamma which is to be
seen here & now, immediate, inviting verification, pertinent, to be
realized by the observant for themselves.
Anekajātisaṃsāraṃ sandhāvissaṃ anibbisaṃ gahakārakaṃ gavesanto dukkhājātipunappunaṃ is about the mind cycling/spinning in "births". The word "birth" ("jati") means "identity", which is a fixed view of self or "beings" ("satta") born from attachment & craving. The "builder of the house" ("gahakārakaṃ") is craving; is all twelve conditions of dependent origination.
Therefore, in terms of Dependent Origination, "identity" ("birth") means things attached to called "acquisitions" ("upadhi"), as follows:
And what may be said to be subject to birth? Spouses & children are
subject to birth. Men & women slaves... goats & sheep... fowl &
pigs... elephants, cattle, horses, & mares... gold & silver are
subject to birth. Subject to birth are these acquisitions, and one who
is tied to them, infatuated with them, who has totally fallen for
them, being subject to birth, seeks what is likewise subject to birth.
"Identity" includes attachments such as "myself", "my life", "my mother", "my father", "my sister", "my daughter", "my brother", "my son", "my friend", "my car", "my computer", "my money", etc.
When these attachments are lost because of impermanence, the idea of "death" arises, such as the "death of my mother", "death of my father", "death of my son", "death of myself".
Because of this "death", there is sorrow, lamentation, pain, grief & despair and suffering.
Therefore, when Buddha destroyed the house-builder of craving & dependent origination, the Buddha also destroyed "identity" or "self". When there is no "identity", there is no "death" and no suffering.
The scriptures say:
Bhikkhu, ‘I am’ is a conceiving; ‘I am this’ is a conceiving; ‘I
shall be’ is a conceiving; ‘I shall not be’ is a conceiving; ‘I
shall be possessed of form’ is a conceiving; ‘I shall be formless’ is
a conceiving; ‘I shall be percipient’ is a conceiving; ‘I shall be
non-percipient’ is a conceiving; ‘I shall be
neither-percipient-nor-non-percipient’ is a conceiving. Conceiving is
a disease, conceiving is a tumour, conceiving is a dart. By
overcoming all conceivings, bhikkhu, one is called a sage at peace.
And the sage at peace is not born, does not age, does not die; he is
not shaken and does not yearn. For there is nothing present in him by
which he might be born. Not being born, how could he age? Not ageing,
how could he die? Not dying, how could he be shaken? Not being shaken,
why should he yearn?