Nirvana is the extinction of rebirth, and birth is its arising. Right?
Is the debate throughout Buddhism on the "difference" between samsara and nirvana one of how to explain, rather than describe or reach etc., the two? So that when we read the Buddha say the two are the same, that means that they are explained in the same way, not that they appear to be the same thing.
From the beginning of the wikipedia article
In the Buddhist tradition, Nirvana has commonly been interpreted as the extinction of the "three fires", or "three poisons", passion (raga), aversion (dvesha) and ignorance (moha or avidyā). When these fires are extinguished, release from the cycle of rebirth (saṃsāra) is attained.
I believe all extant Buddhist traditions, including theravada, believe that we are reborn from moment to moment: a lifetime as much as the aggregates at an instant. Also, see 'dependent origination'.
Both the Sarvāstivāda [the Mahayana's tripitaka] and the post-canonical Theravāda constructed a radical doctrine of momentariness (Skt., kṣāṇavāda, Pali, khāṇavāda) that atomizes phenomena temporally by dissecting them into a succession of discrete, momentary events that pass out of existence as soon as they have originated
This is true of the Abhidhamma, see Karunadasa:
in the Pali Suttas, unlike the Abhidhamma, the notion of change is not presented either as the doctrine of momentariness or as a formulated theory of moments... the Pali suttas say that it is peculiar to the Abhidhamma.
And this from Charles Bartley
After the death of an enlightened one there is no rebirth
Or this from Keown
At the age of 80 he passed away into final nirvana, from which he would not be reborn
Moreover, non-abiding is likewise the end of rebirth. Tharpa:
Non-abdiding in nirvana in the irreversible cessation... of all rebirth
The Buddhists personalists -- which no longer exist and shared in much the same sutta basket -- believed not just that only a sentient being could create -- in series -- a new mind-body after death -- all Buddhists claim this -- but that something was the same during its life.
it is the pudgala that appropriates and sustains a body for a certain amount of time.