I think that "attachment" is as explained in these topics:
I think that "detachment" is the non-doing (or un-doing) of attachment ... e.g. by removing the conditions in which attachment arises.
One of the definitions of how attachment arises is as described by the 12 nidanas, a.k.a. "dependent origination". I think that the doctrine (dhamma) teaches that a way to interfere with this "cycle of becoming" is a technique called "guarding the senses", which is e.g. described here.
IMO "mindfulness" in this context could include:
- Remembering to guard the senses (mindful awareness of the senses)
- And, remembering why you guard the senses (remembering the dhamma)
- And, the resulting state of mind (e.g. not getting carried away on a flood)
Mindfulness is (more generally) sati i.e. one of the divisions of the noble eightfold path -- see e.g. the Sati (Buddhism) article on Wikipedia.
I think that (perhaps confusingly) the English word "mindfulness" is also used in a non-Buddhist context. The Sati (Buddhism) article says, in its prologue ...
This article is about Buddhist mindfulness. For information on mindfulness in psychology, see mindfulness.
... which ("mindfulness in psychology" i.e. not Buddhist) was the article you cited in the OP.
My (very limited) experience has been that it's not easy to explain Buddhist dhamma (to people who aren't already interested in it as such) -- see How to explain what Buddhism is?
When "mindfulness" is taught in the West (e.g. here) then that may be without mentioning Buddhism, or the Four Noble Truths, or anything like that.
I've no experience with Mindfulness-based stress reduction so I can't talk about it. Perhaps (I don't know) it's similar to some techniques or thought-processes described in the answers to Experiencing physical pain (perhaps even including some awareness of impermanence of sensation, but basically stripped of any overtly Buddhist references ... and of any 'foreign languages' e.g. words like sati).
One topic I have a little second-hand experience of is "mindfulness" as a therapy of Post-traumatic stress disorder (PTSD) -- as follows. The way that might work is, imagine you're attacked one day -- so someone cuts an artery, your blood sprayed across the walls and so on. Years, decades later, you still think of that incident ... and have "panic attacks", and flashbacks:
The term is used particularly when the memory is recalled involuntarily, and/or when it is so intense that the person "relives" the experience, unable to fully recognize it as memory and not something that is happening in "real time".
Mindfulness might teach you to be aware of what's happening now, what actually you're sensing now -- when you're taking a shower, for example, the noise and feel of the water, the sight of clean (bloodless) bathroom tiles, the knowledge (thought) of having locked doors and no-one but friends around and so on -- as an antidote to, a replacement for, a detachment from the flashback you were previously experiencing.
I think that's worthwhile, if it's effective (to relieve suffering caused by being attached to some traumatic past experience).
I think you might see or say, though, that that's not quite like Buddhist mindfulness -- which if anything might be trying to keep you from becoming overly emotionally invested in present sensations -- and the desires (or cravings), not just aversions, occasioned by past experiences.
Reading the question again, I see you wrote ...
a state in which a person overcomes his or her attachment to desire for things, people or concepts of the world
... which I read as "detaching from views" (and so, perhaps including "self-views" for example)..
I think Andrei often mentions something like that, e.g. in this answer:
While this may partially exist before wisdom, only complete realization of Emptiness gives unlimited insight into other perspectives. Normally, this ability is hindered by ego's attachments, biases, fixed views etc. The enlightened mind is completely free from these and can truly see from all perspectives, without distortion.
I'm inclined to agree, but I don't know many references in the Pali canon to support this. Instead a lot of the suttas (and the beginning of the Noble Eightfold Path) concentrate on distinguishing "right view" from "wrong view".
Some of the examples of suttas which may teach "detachment from concepts" include the simile of "dhamma as a raft", and the simile of the "blind sectarians and the elephant".