Do Mahayana Buddhists explain birth and extinction in the same way?

I know really little of Mahayana doctrine, sorry.

For a start some people talk about "perception" rather than "appearance".

The "appearance" of a thing sounds like it's some inherent property of that thing -- which is contradicted by the doctrine on emptiness, isn't it?

Whereas "perception" talks about how (and/or whether) you perceive it -- if "appearance" sounds like it's "objective" then "perception" sounds more "subjective".

And I don't know but I guess that "nirvana is samsara" is a statement about perception -- that "nirvana" is a dualistic and value-laden statement, and therefore etc.

And is the vedanā doctrine related to that, or is that quite different?

But anyway that is (about "perception"), as well as possibly about "appearance" -- which I guess means (in a Mahayana context), "where do you draw boundaries which distinguish one thing from another?"

And the first part of the question asked about the difference between "explanation" and "description" -- but I guess I see no practical or significant difference between those two words, so, no go there.

You mentioned "reaching" too.

That might be a temporal statement. For example SN 56.11 talks about things which are "to be done" (in future), "are done" (presently), and "have been done" (see also "Perfect (grammar)").

The suttas talk of a gradual training (and step-by-step explanations).

I think that ("gradual") is true of Tibetan Buddhism too -- but it can be a subject of debate, so maybe different schools etc.

There's some Buddha-nature doctrine: I think one might consider whether that means that things 'can' have -- or whether they 'already' have -- that nature.

You might see also this answer for one description (from Shin) -- possibly the Brahmana Sutta (about "going to the park") from the Pali suttas.

As for this ...

I believe all extant Buddhist traditions, including theravada, believe that we are reborn from moment to moment

... then, "maybe yes, something like that ..." -- but that's possibly not the only truth. I think that traditional Buddhists see it (whatever it is) as being true over several time-scales -- moment-to-moment, life-to-life, even aeon-to-aeon.

Perhaps one (educated) view is that if rebirth is moment-to-moment, then "death" too is something of a fiction. Something like this answer -- What's the value or harm of a literal belief in rebirth? -- which I think is Theravada, might be orthodox even if some people concentrate more or only on what's "evident" etc. I don't much want to get into describing what or who is reborn. I think that the Pali suttas suggests that's "only suffering" -- Vajira Sutta. Also "identity views" or "self-views" (including "I do or I don't exist") are an origin of suffering, and a thicket of views and a product of unwise attention. Whether and when the Tathagata "exists" is famously "undeclared".

And that too might be "gradual", at least according to the Theravada doctrine, see e.g. How are 'conceit' and 'identity-view' not the same?

In Mahayana (and perhaps before) the existence or body of the Buddha came to be associated with the Dharmakāya.

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:

17. 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.
18. 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.
19. There is no distinction whatsoever between samsara and nirvana; and there is no distinction whatsoever between nirvana and samsara.
20. 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.