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?