Nanamoli/Bodhi in their Majjhima Nikaya translation (4th ed) writes the following note associated with "experiencing the whole body" as present in the Sathipattana Sutta:
The Majjhima Nikaya Atthakatha explains "experiencing the whole body" (sabbakaya patisamvedi) as signifying that the meditator becomes aware of each in-breath and out-breath through its three phases of beginning, middle and end. In the first edition I followed this explanation and added in brackets "of breath" after "whole body". In retrospect, however, this interpretation seems forced, and I now prefer to take the phrase quite literaly. It is also difficult to see how patisamvedi could mean "is aware of", as it is based on a verb "to experience".
Here is what Thanissaro Bihkkhu wrote on a footnote for the Anapanasati Sutta:
The commentaries insist that "body" here means the breath, but this is unlikely in this context, for the next step — without further explanation — refers to the breath as "bodily fabrication." If the Buddha were using two different terms to refer to the breath in such close proximity, he would have been careful to signal that he was redefining his terms (as he does below, when explaining that the first four steps in breath meditation correspond to the practice of focusing on the body in and of itself as a frame of reference). The step of breathing in and out sensitive to the entire body relates to the many similes in the suttas depicting jhana as a state of whole-body awareness (see MN 119).
In The Experience of Samadhi, Richard Shankman interviews Ajaan Thanissaro:
RS: In the Satipatthana Sutta, the anapanasati Sutta and the Kayagatasati Sutta, one is instructed to breath in and out, experiencing the whole body. Some interpret that it is talking about experiencing the whole physical body breathing, while others say it means staying with the whole duration of the breath but the focus can be at any point.
AT: It's the whole body.
RS: Does it matter?
AT: One of the drawbacks of concentration that's too one-pointed is that you're blocking out many areas of your experience which means that a lot of things can hide away in the areas you're blocking out. If, however, you develop more of a 360-degree awareness of the body, you're more likely to be conscious of more peripheral events in the mind. Also, if awareness is a whole-body awareness, it's a lot easier to maintain the state of concentration as you open your eyes and move around. [...]. If you only have one point that you are totally focused on, then as soon as you move from that point, your concentration is destroyed. But if you've got the whole body as your framework and you're constantly mindful of this framework, events can come through and go out, leaving the framework undisturbed.
Bhante Gunaratana, on his interview:
The Buddha has said that breath alone is one of the bodies. What do we have in a body? We have four elements: earth, air, water, fire. This is what we find in the breath. And therefore, the breath alone is fully qualified to be called a body, the breath body.
Therefore, if one practices the first tetrad, that individual has practiced mindfulness of the body just by practicing mindfulness of the breath. Some people ignore this particular sentence where the Buddha says the breath is a body, and they give their own interpretation.
Ajhan Brahmavamso, on his interview:
The word kaya in Pali is well translated by the English word "body". We have a physical body, but also have a body of evidence, a body of experience, and these are all immaterial "bodies". We use the word "physical body" just to make sure we know what body we are talking about. kaya, just like the word "body", means an accumulation of things, a conglomerate, a group of things.
Pervading the "kaya" with the jhana factors means that the experence of bliss pervades the whole "mental body" throughout the jhana. It does not mean experiencing the jhana with the physical body. That is a misunderstanding of the use of the word kaya in Pali. It is quite clear that in jhanas you cannot feel anything to do with the body (*).
In the same book, Richard Shankman devotes an appendix titled "Does it matter where you watch the breath?", making further considerations.
Also, Shaila Catherine, in her Focused and Fearless, seems to teach that step as focusing on the breath "as an object", rather than focusing on the specific sensation of in and out breath. Its hard to say from what lineage she got this, as she trained with monks from many different schools (maybe Ven. Pa-Auk Sayadaw?).
Ajahn Brahm, also wrote on this matter:
The third step is called in Pali sabba-kaya-patisamvedi, experiencing the whole process of breathing. A minority of teachers mistake the Pali term kaya to mean your physical body and so wrongly assume that now you are meant to direct your attention onto all the sensations in the whole of your physical body. This is an error. The Buddha clearly stated in the Anapanasati Sutta that he regarded the process of breathing as "a certain body (kaya) among the bodies". Moreover, the direction of the first twelve steps of anapanasati is toward simplifying the object of awareness, not making it more complex. thus, this third step is where your mindfulness increases its agility sufficient to observe every sensation involved in the process of breathing.
-- Mindfulness, Bliss and Beyond - A meditator's handbook
On a related issue, Joseph Goldstein notes in his book:
At this point in the sutta, there is an interesting change of language. As Analayo points out, in the first two exercises, the Buddha uses the verb "to know": breathing in, one knows one is breathing in, etc. But in the next two exercises of mindfulness of the breath, the Buddha uses the verb "to train". [...] This shift of language [...] suggests an increasing level of intentionality in our practice as we broaden our awareness from the breath to the whole body.
-- Mindfulness: A Practical Guide to Awakening
And then he goes on to summarize the conflicting views of "whole body", but doesn't really take any position, just sees them as useful alternatives.
Finally, Analayo writes on his PhD-thesis-turned-book on the Satipatthana sutta:
The third and fourth steps of mindfulness of breathing, alike in both the Anapanasati Sutta and the Satipatthana Sutta, are concerned with experiencing the "whole body" (sabbakaya) and with calming the "bodily formation" (kayasaukhara). In the present context, the "whole body" can be taken literally to refer to the whole physical body. Understood in this way, the instruction points to a broadening of awareness, a shift from the breath alone to its effect on the en tire body. According to the commentaries, however, the "whole body" should be understood to refer, more figuratively, to the "body" of the breath. By understanding the "whole body" as the whole breath-body the instruction then indicates full awareness of the beginning, middle, and end stages of each breath. This interpretation can claim support from the same Anapanasati Sutta, since the Buddha here identified the breath as a "body" (kaya) among bodies. An argument against this interpretation, however, could be that the cultivation of full awareness of the length of the breath was the task of the previous two steps, knowing a long or a short breath, which already required the meditator to be aware of each breath from beginning to end. One would therefore expect this next step in the progression to introduce a distinctly new feature for contemplation, such as, for example, a shift of awareness to include the whole physical body.
-- Satipaṭṭhāna, the Direct Path to Realization
(*) Other interviews present conflicting declarations with that assertion made by Ajhan Brahmavamso.
Edit: included texts by Ajahan Braham, Shaila Catherine and Joseph Goldstein