Which Buddhists denominations agree with the difference between pain and suffering as in many DBT texts? Which disagree?

Buddhism views suffering as coming from 8 different forms, as below:

Suffering of birth;

Suffering of old age,

Suffering of sickness,

Suffering of death,

Suffering of getting what you don't want,

Suffering of not getting what you want,

Suffering of getting what you want while not being able to hold onto it forever,

and then there is the most mysterious one of all, which is all pervasive suffering or misery.

Therefore, in mainstream Buddhism, it is widely accepted that "Suffering is inevitable" so long as your are still in the wheel of re-incarnation. (Un-enlightened). The core of Buddhism teaches about treating suffering (4 noble truths) through enlightenment (8-folds path). Hence, from the above list, there was no further dissection of pain vs suffering, because that would be redundant and further complicate matters.

In comparison, it seems DBT basically replaced the word "Suffering" from Buddhism's context, with the word "Pain" in DBT's context. Not saying DBT is less comprehensive, after all it is a researched therapy with Buddhism flavors.

I believe it's probably more useful and easier to be accepted to treat those people with different conventional/religious backgrounds.

Always be mindful that - Suffering and pleasure are two sides of the same coin.

Does mainstream Buddhism agree with the above?

Yes I think so.

I don't know any DBT except what you quoted, so by "the above" I assume you mean the first quote, about "negative emotions" and "finger-trap".

The "negative emotions" are a subject of Buddhism, see e.g. Kleshas (Buddhism) where they're translated as "Afflictive emotions". The different schools enumerate them differently (depending on how you analyse them there may be 2, 3, 6, or 50 of them), but I think that all schools agree they exist and are a problem.

Some schools might suggest there's only one way to solve the problem -- i.e. "the Noble Eightfold Path" (which is the fourth of the "Four Noble Truths") -- or perhaps all schools agree with that. I think that user13190's answer is a summary or paraphrase of the Dhammacakkappavattana Sutta.

Even so, different schools have different practices and so on.

When you get into details of how to solve the problem, what the solution is, I suspect that Buddhism probably differs from DBT -- otherwise wouldn't DBT be Buddhism, indistinguishable?

Assuming you may find the Dhammacakkappavattana Sutta difficult, another popular snippet might be this from the Dhammapada:

3. "He abused me, he struck me, he overpowered me, he robbed me." Those who harbor such thoughts do not still their hatred.

4. "He abused me, he struck me, he overpowered me, he robbed me." Those who do not harbor such thoughts still their hatred.

5. Hatred is never appeased by hatred in this world. By non-hatred alone is hatred appeased. This is a law eternal.

I think that (i.e. whether an afflictive emotion like "hatred" is "stilled" or not) is reminiscent of the "finger-trap" you mentioned.

The first two verses:

1. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with an impure mind a person speaks or acts suffering follows him like the wheel that follows the foot of the ox.

2. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness follows him like his never-departing shadow.

... imply that the mind (e.g. it whether it harbors or doesn't harbor such thoughts) conditions (creates conditions for, is followed by) suffering or happiness -- maybe (I don't know, but I guess) DBT says something along those lines.

Oh -- the title of what you quoted, "acceptance and change", sound a bit like Buddhist doctrine too -- Buddhism teaches that things are impermanent (change), and I think it teaches (e.g. in the second of the four noble truths) that being attached to things is a cause of suffering -- see also for example Why do the Noble Truths talk about 'craving', instead of about 'attachment'?

Buddhism is kind of complicated though; people might resist it for various reasons, including the one you mentioned -- or for an opposite reason, e.g. that people don't want to see "suffering".

Also I might have given have an opposite reply, and said that everything is suffering (or is, at least, unsatisfactory) -- sabbe sankhara dukkha -- that too is hard to explain, and even that is a topic which different schools of Buddhism may summarise differently, see e.g. Differences between the tilakkhana and the Dhamma seal

It's not clear what you mean by "pain" in the question.

I assume that maybe you don't mean physical pain -- see e.g. Experiencing physical pain -- so maybe you mean something metaphorical.

In that case you'd asking whether there's a difference between "metaphorical pain" and "suffering", which I think is quite vague.

The Pali word that's translated as "suffering" is Dukkha (maybe we should define the dukkha and suffering tags to be synonyms).

And actually dukkha doesn't mean exactly the same as suffering (so much so that people prefer to refer to it as "dukkha" instead of as "suffering"), so that's another way in which the question is inherently imprecise.

Reading the quote "... a type of psychotherapy that combines behavioral science with Buddhist concepts like acceptance and mindfulness" reminds that I have had one second-hand (i.e. hearsay) experience of a DBT practitioner: which was, as a treatment for PTSD (the trauma of having been assaulted), being advised to be "mindful" of the present environment. For example, when you're taking a shower, instead of having "flashbacks" (i.e. reliving the experience of being assaulted), to "remember" where you are at the moment, e.g. in the shower, feeling the touch of water, hearing the sound of water, and so on.

That is not unlike some practice Buddhist practices, where "mindfulness" is a meditation and or part of the doctrine -- see e.g. What does sati mean? but see also Where is the suttas is 'sati' defined as present moment awareness?

OP: 'Pain is inevitable. Suffering is optional.'

I am quoting the viewpoint of the Theravada school which agrees with this statement, and I've never heard of another school disagreeing with this statement.

According to Itivuttaka 44, enlightened ones can still feel pain and pleasure:

"What, bhikkhus, is the Nibbana-element with residue left? Here a bhikkhu is an arahant, one whose taints are destroyed, the holy life fulfilled, who has done what had to be done, laid down the burden, attained the goal, destroyed the fetters of being, completely released through final knowledge. However, his five sense faculties remain unimpaired, by which he still experiences what is agreeable and disagreeable and feels pleasure and pain. It is the extinction of attachment, hate, and delusion in him that is called the Nibbana-element with residue left.

However, the enlightened ones don't suffer, even when they feel pain or are deprived of pleasure (from DN16):

And soon after the Blessed One had eaten the meal provided by Cunda the metalworker, a dire sickness fell upon him, even dysentery, and he suffered sharp and deadly pains. But the Blessed One endured them mindfully, clearly comprehending and unperturbed.

Also see this answer.

In the Nakulapita Sutta:

"So it is, householder. So it is. The body is afflicted, weak, & encumbered. For who, looking after this body, would claim even a moment of true health, except through sheer foolishness? So you should train yourself: 'Even though I may be afflicted in body, my mind will be unafflicted.' That is how you should train yourself."

Generally speaking, "Buddhism is like Christianity" - in the sense that it has many different schools and sects that have their own practices and disagree on interpretations.

However, when it comes to suffering vs pain, most Buddhist nominations should agree with that statement. In fact I suspect this entire idea was taken by DBT from Buddhism. Suffering is not the best word for dukkha though, hence your confusion/resistance. Dukkha is the painful feeling we have when things are wrong and we can't do much about it.

If that makes you feel better, here is the same phrase in some alternative renditions:

'Pain is inevitable. Unhappiness is optional.'

'Pain is a part of life. Frustration is up to you.'

The idea is that external circumstances are not in our control, even our state of mind is not always in our control... but our attitude - our deep inner peace - is entirely up to us. Even if we're in pain and dying, we can be at peace an die gracefully, can't we? Let alone when we have normal difficulties.

This idea is slightly more palatable than telling someone their suffering is self-inflicted, but even this may be hard to accept for someone used to being in the victim consciousness. We're very used to explaining our unhappiness in terms of the circumstances. But as my Zen Master used to say, keeping our mind clean from unhappiness is like keeping our kitchen clean from the c**kroaches. It is entirely our responsibility. In fact, rather than feeling pity for ourselves for being unhappy - we should be ashamed of not keeping our "kitchen" clean.

This is Emotional Intelligence 101, and a lot of it comes from Buddhism. Consider the Four Right Efforts:

... monks, you should apply effort toward:
- non-arising of bad, pathological states of mind that have not yet arisen.
- abandonment of bad, pathological states of mind that have arisen.
- arising of wholesome states of mind that have not yet arisen.
- maintenance and development of wholesome states of mind that have arisen.

Even the Noble Truth itself is a form of the same principle. The core idea is that dukkha (unhappiness, frustration) comes from an inner conflict. Conflict between how things "should be" and how they "are" - both sides modeled by our mind. The Buddhist path leads a person to the ultimate state of peace called "suchness" (tathata) - the state when "is" and "should" is no longer in conflict. This is achieved through a combination of objective (behavioral) changes in one's lifestyle and subjective (attitude) changes in one's psyche.

Buddhist Nirvana is the culmination of this path from conflict to suchness.