What puts out the mind

As to the question "die in the sense atheist attaches to death". The answer is no .i.e. complete extinction, total annihilation, because there is the unborn which is deathless.

"Death has no (ultimate) reality.".. is the correct view ("who" dies?).

Suppose we call "extinction" (by which I mean nirvana or final nirvana) whatever puts out the wandering mind / mental states.

I think the first paragraph, "extinction or nirvana is whatever puts out the mind", suggests that nirvana has been reified and is seen as an agent which acts to put out the mind.

FWIW I don't know that it's that: I see it as a description of a state (perhaps a state of mind), not an agent: it's the state-of-being-extinguished (or of having-been-extinguished), not the agent-which-does-the-extinguishing. It's like the difference between Enlightenment (which is a noun) and enlightened (which is an adjective).

Also the word "extinguish" comes I think from the analogy of a flame: if you look at it, a flame appears to cling to what it's burning ... if it's burning a match or a log, the flame kind of clings to and dances around the wood.

I think that the "nirvana is an extinguishing" analogy comes from saying, "what would happen to a flame if it stopped clinging to its fuel source?"

So if you're asking "what puts out the mind" then that's like asking "how do you attain nirvana or become enlightened", and I think that the answer to that is "Buddhist practice" e.g. the Noble Eightfold Path (or something else though maybe perhaps similar in later versions of Buddhism).

you might like to add extinction of greed, hate and delusion, which are the fuel. :)

In case it would help you to know, there's some debate about "what is the fuel?" in the analogy.

Wikipedia's Nirvana with and without remainder of fuel references Gombridge. On page 66 of Gombridge's How Buddhism Began (in a chapter titled "Metaphor, allegory, satire") he claims that:

• He showed in an earlier paper of his that the "three fires" was a reference to the (literally) three fires which a brahmin householder was expected to keep alight.
• Later generations of Buddhists weren't interested in debates with Vedic brahmans and forgot the origin of the metaphor.
• In later (e.g. Mahayana) times the "three fires" became associated with the "three poisons".

You can maybe kind of see that:

• Wikipedia's Three fires (Buddhism) link redirects to the three poisons
• The comments/conversation underneath this answer suggest that the "three poisons" are more central in a later version of Buddhism than an earlier version.

In contrast, the following suggest that it's the khandhas which are the fuel:

• In Five Piles of Bricks – The Khandhas as Burden & Path Thanissaro Bhikkhu writes,

  The Buddha introduced the concept of the khandhas in his first sermon in response to the first of these questions. His short definition of suffering was "the five clinging-khandhas." This fairly cryptic phrase can be fleshed out by drawing on other passages in the canon.

  The five khandhas are bundles or piles of form, feeling, perception, fabrications, and consciousness. None of the texts explain why the Buddha used the word khandha to describe these things. The meaning of "tree trunk" may be relevant to the pervasive fire imagery in the canon — nibbana being extinguishing of the fires of passion, aversion, and delusion — but none of the texts explicitly make this connection.

• In The Fire Sermon the Buddha asks, "What is burning?" and answers, that sense organs (e.g. eye), form, sense-contact, sense-consciousness, etc., are burning with the fires of lust and hate and delusion.

So apparently the three poisons are the three fires, but the khandhas (to which these fires cling) are the fuel.

is nirvana categorically not an agent?

Well, who knows, right? That's how I categorized it. If other people think I'm wrong then hopefully they won't let me mislead you, and will instead downvote this answer, or post a comment or post an alternative answer (currently it appears that two people agreed with it). Some people didn't understand your question though, which might be why it doesn't have more answers.

FWIW I think that maybe later schools of Buddhism might 'reify' more:

• Later schools might (I don't know) see "Buddha-nature" as an agent.

  Buried in Wikipedia's Buddha-nature article is the following quote,

  The Lankavatara-sutra contains tathagata-garba thought, but also warns against reification of the idea of Buddha-nature, and presents it as an aid to attaining awakening:

  Is not this Tathagata-garbha taught by the Blessed One the same as the ego-substance taught by the philosophers? The ego as taught by the philosophers is an eternal creator, unqualified, omnipresent, and imperishable.

  The Blessed One replied: [...] it is emptiness, reality-limit, Nirvana, being unborn, unqualified, and devoid of will-effort; the reason why the Tathagatas [...] teach the doctrine pointing to the Tathagata-garba is to make the ignorant cast aside their fear when they listen to the teaching of egolessness and to have them realise the state of non-discrimination and imagelessness

  Would you say that "egolessness, non-discrimination and imagelessness" can be "categorized as an agent"? I'd say no, that's not an agent. That might be splitting hairs though, because although not an "agent" I could call it a "condition". The theory of Dependent Origination (and the second noble truth) emphasizes that things are conditioned/conditional/dependent, they arise or don't arise depending on certain conditions. Perhaps the condition could be called an "agent", if it causes or allows something, but maybe it's not "acting" or "taking an active role", not a doer of an action, not a subject of active verb, it is "devoid of will-effort".

• I think that, certainly, some view Bodhisattvas as agents.

• There are other 'states' which I view as being more 'active' than nirvana (although they're still just 'states' and not 'agents' themselves): i.e. the brahmavihāras (or appamaññā) i.e. the states like 'metta' and so on.

• Gombridge also claims that nirvana is viewed differently between Mahayana versus earlier schools.

  I hope it is not too farfetched to suggest that this may have contributed to an important development in the Mahayana: that it came to separate nirvana from bodhi, ‘awakening’ to the truth, Enlightenment, and to put a lower value on the former (Gombrich, 1992d). Originally nirvana and bodhi refer to the same thing; they merely use different metaphors for the experience. But the Mahayana tradition separated them and considered that nirvana referred only to the extinction of craving (= passion and hatred), with the resultant escape from the cycle of rebirth. This interpretation ignores the third fire, delusion: the extinction of delusion is of course in the early texts identical with what can be positively expressed as gnosis, Enlightenment.

After all that, it's still difficult to be sure of what nirvana is w.r.t. death. Wikipedia summarizes it as follows (and includes a lot of quotes to support that summary),

What happens with one who has reached nirvana after death is an unanswerable question. The five aggregates vanish, but there does not remain a mere "nothingness."

*... puts out the wandering mind / mental states.

And even apart from whether it's an agent, I don't know that nirvana (assuming you attain nirvana during your life, which is the only time you can attain it) "puts out the wandering mind / mental states".

The bit I mentioned earlier, above, about fire, suggests that nirvana puts out clinging (or craving), and/or that putting out of craving results in nirvana.

The mind continues (that's part of the "burden" that's talked about, which people put down when they die). Whether you'd call it a "wandering" mind I'm not sure. Ability to concentrate (right mindfulness, right concentration) are part of the eightfold path, so maybe not "wandering".

On the other hand some people (even before nirvana) are said to acquire unusual types of special knowledge so in a way perhaps you could say that their mind is able "wander" even further than usual (e.g. to see things far away, to view past lives, etc.) – but maybe just this last paragraph is "Buddhist doctrine" and contradicts what you asked for i.e. an "atheistic" view.