How can mindfulness be about immovable attention? There seem to be a lot of disagreements about the very basic term "mindfulness". Here is one of them.

Yesterday I listened to a talk, where it was it was said that if one achieve total concentration on an object, there is no mindfulness. This makes sense to me, as ALL attention is on the subject and none is left to be mindful of what is going on (it was also mentioned that there is no wisdom gained from this, as it is from the moving, momentary concentration in mindfulness that one makes realizations about the four noble truths).  This is all in accordance with my understanding of the difference between concentration and mindfulness.

Then later at the evening I read an article by Ajahn Brahmavamso ("The quality of mindfulness") where he says that "with development, you can experience immovable mindfulness. The mindfulness that is on one thing entirely...The Buddha said this reaches its peak in the fourth Jhanas". I have read and heard a thousand other places that this is what he said about samadhi (concentration). And that one of Buddhas main contributions was the introduction of sati/mindfulness which among other things makes one aware of impermanence.  How should "immovable mindfulness" ever give any realization of impermanence...immovable means permanent.  He also says that "Once you know that type of mindfulness, then you know how ridiculous it is to think you can become Enlightened without Jhanas. Without such powerful mindfulness you can't get the powerful insights". To my knowledge Jhanas and one-pointedness (which I would call concentration) is not about insight. To my knowledge Buddha, before he found the path to (his) enlightenment,  was taught meditation techniques which aimed towards being completely absorbed, but that he found that this did not bring him what he searched for. Isnt this one of the major points in buddhist teaching?
When one gets absorbed, focused, or mindful in anything, no matter what the object is, the mental factor called “ekagghatha” takes over and make the mind latched on to it. Being mindful depends on the situation. The kind of mindfulness one has when driving a car is different from the mindfulness when reading a book, and the mindfulness needed to attain a jhana is different from those two.  

The jhana that one comes to by using the breath only would not help in walking this Path. As you correctly pointed out Buddha figured this out, and managed to break through and achieve Nibbhana. Just by focusing the mind on the breath, one can achieve only a mundane type of jhana. Also trying to be mindful by doing “walking meditation” by saying “lifting the foot”, “putting it down” would only get you to this lower level mundane jhanas.

The higher levels of Jhana that furthers the Noble Eightfold Path are achieved by supermundane “ekagghatha”  that one comes to by coming to upacara samadhi.  When listening to a Dhamma discourse or reading about a Dhamma concept one can get to this type of mindfulness.

When contemplating on bad consequences of mind-made pleasures, willingly giving up by comprehending their bad outcomes one is able to come to the higher levels of Jhana.

If one’s intention is to walk this Path, the focus should be something related to Nibbana. It could be a Dhamma concept or Nibbana itself. Focusing on the peacefulness of discarding sense pleasures can be done only when one comprehends the true meaning of anicca, dukkha, anatta. By being mindful the correct way, one can come to samma sati.