Do Mahayana Buddhists explain birth and extinction in the same way? 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.
There is no difference in manner or extent between the rebirth we experience from moment to moment and the rebirth we experience from life to life. To misunderstand this is to misunderstand the subtlest meaning of the Buddha's teachings on emptiness or anatta.

Therefore, think about what this means for your question. When an enlightened one first becomes enlightened and reaches nirvana does an enlightened one continue to exist from moment to moment? Does he not exist from moment to moment? Both exists and does not exist? Neither exists nor not exists? Whatever the answer... then to that exact same extent and manner an enlightened one will continue to exist from life to life. To say otherwise is to assert that something has materially changed such that after nirvana an enlightened one continues in that very life, but utterly ends or is annihilated at the end of that life. This cannot be the case. It would mean an enlightened one substantially existed or inherently existed in some way and that this ended or was annihilated at death.

It is important to know the manner in which the three poisons end. The real liberation comes through knowledge or the extinction of ignorance. From this, the other poisons are extinguished as they rely upon ignorance to sustain them. Nirvana is nothing more than waking up to the knowledge of reality as it really is rather than by perceptions and conceptions clouded by ignorance. 

Think about what Nagarjuna is saying here:


  
  It is not asserted that the Blessed One exists after his passing away; nor is it asserted that he does not exist, both exists and does
  not exist, or neither exists nor does not exist.
  Even while he is living, it is not asserted that the Blessed One exists; nor is it asserted that he does not exist, both exists and
  does not exist, or neither exists nor does not exist.
  There is no distinction whatsoever between samsara and nirvana; and there is no distinction whatsoever between nirvana and samsara.
  The limit of nirvana and the limit of samsara: one cannot even find the slightest difference between them.
  


Verse 17 and 18 assert that there is no distinction between the way an enlightened one exists from moment to moment as compared to life to life.

Verse 19 and 20 draw on the prior verses to show there is no distinction between nirvana and samsara. Nothing has materially changed except where before there was ignorance now there is knowledge.

It is like waking up from a dream even as the dream perceptions continue... there is no longer any doubt that one is perceiving a dream. Not one iota of doubt. The ignorance of the dreamer has been replaced with the sure knowledge of an awakened one.