The methods:
Ajahn Brahm:
He describes the words 'breathing in-out sensitive to the entire body' and 'breathing in-out calming the body fabrications' in this book. He says that here is meant with the body the breath-body and not the entire body. He gives for explanations that the Buddha called the breath a body among bodies and that in the first couple steps of Anapanasati the object of meditation (the breath) is simplified (from knowing the breath as long or short to just knowing you are breathing) and that it would be illogical to make it more complicated in step 3 or 4 in the Anapanasati to suddenly be sensitive to the entire body. He also explains that 'calming bodily fabrications' means calming the breath-body, and the rest of the body will follow.
My opinion:

Ajahn Brahm has quoted the sutta correctly but translates & interprets it incorrectly. Only Ajahn Buddhadasa has translated & explained it correctly. The Pali is 'sabbe kaya', which means '
all bodies'. The word 'sabbe' means 'all', such as in the phrases: "Sabbe satta sukita hontu" and "sabbe dhamma anatta" (may all beings be happy; all things are not-self). 'All bodies' refers to the breath as a body & the physical body as a body. It can also refer to the mind as a body (kaya), such as in the term 'nama kaya'.
The Pali 'sankhara' does not necessarily mean 'fabrication'. In fact, in this context, it means '
fabricator'. Therefore, the breathing in & out is the body fabricator because it fabricates or conditions the physical body.
For example, when the breath is long, fine, smooth & calm, the physical body will feel at ease. When the breath is coarse, short, rough & agitated, the physical body will feel uncomfortable & distressed.
At the 3rd step, what should be comprehended is the relationship between the 3 kaya. Insight (vipassana) should occur here into
cause & effect. What should be comprehended is the relationship between the mind, the breath & the body, which is:
"When the mind is like this, the breath becomes like this & the body becomes like this. When the mind is like that, the breath becomes like that & the body becomes like that".
This is the meaning of the 3rd step of experiencing all bodies (sabbe kaya).
Thanissaro Bhikku:
He is talking about the whole body. So 'breathing in-out sensitive to the entire body' and 'breathing in-out calming the body fabrications' means sensitive to the entire body, and not only the breath. 'Calming bodily fabrications' means literally calming all of the bodily stress-points and be sensitive to the entire body. This method also includes visualizing the breath coming out of every pore in the body, and in his book he describes more of these visualizing stuff. I am sceptible about this because I have the feeling visualizing stuff (an example is that he describes visualizing the words from someone else floating around you instead of coming towards you in an aggressive communication) is not experiencing things as they really are.
Best to reflect with reason & intelligence. Can the mind be aware of the whole body, such as the little finger, the little toe & every hair on the head?
However, strangely, Thanissaro Bhikku has discussed the 'kaya sankhara' (body fabricator) correctly in his essay: 'Shape of Suffering', as follows:
2) Fabrication: the process of intentionally shaping states of body and mind.
These processes are of three sorts:
a) bodily fabrication: the in-and-out breath,
b) verbal fabrication: directed thought and evaluation, and
c) mental fabrication: feeling (feeling tones of pleasure, pain, or neither pleasure nor pain) and perception (the mental labels applied to the objects of the senses for the purpose of memory and recognition).
As you walk to the door of your parents’ house, thinking about the situation (2b—verbal fabrication), you pull up memories of things your uncle has done in the past (2c—mental fabrication). This provokes anger, causing your breathing to become labored and tight (2a—bodily fabrication). This makes you uncomfortable (2c—mental fabrication), and you are aware of how uncomfortable you feel (3—consciousness). Hormones are released into your bloodstream (4 f through 4i—Form).
http://www.accesstoinsight.org/lib/authors/thanissaro/shapeofsuffering.pdf
Hopefully we can start a nice discussion here. I use the Ajahn Brahm method at the moment, but I'm trying to find out which one is the 'right' method and what are the pros and cons for each method. I am also very interested in which method you use, and why you think this one is better than the other etc.
In my opinion, Ajahn Brahm has the 'right' method, particularly his emphasis upon letting go as the basis for developing samadhi.
However, only Ajahn Buddhadasa has explained the fruition of practise correctly (
here).
Best wishes

As for the Pali, it is quite clear. The Pali is 'kāyasaṅkhāraṃ', which refers to the 'breathing in & out'.
sikkhati, ‘passambhayaṃ kāyasaṅkhāraṃ passasissāmī’ti sikkhati
He trains himself: calming the body-conditioner I shall breathe in. He trains himself: calming the body-conditioner I shall breathe out.
MN 118
Assāsapassāsā kho, āvuso visākha, kāyasaṅkhāro
Friend Visakha, in-&-out breaths are the bodily saṅkhāro
MN 44