The term ‘Madhyamika’ means ‘Middle Way’, and Madhyamika is a philosophical school of Mahayana Buddhism (the second phase of Buddhism that began about 500 years after the Buddha and spread from India into China, Tibet and Japan). If you see the term ‘Middle Way’ highlighted in a Buddhist context, it is most likely to be refer to the thought and influence of this school, particularly its lead thinker Nagarjuna. Nagarjuna, in texts like the Mulamadhyamakakarika (‘Fundamental Wisdom of the Middle Way’) does a lot to develop and apply Indian Buddhist thought about the Middle Way beyond the base created by the Buddha. It is Mahayana Buddhists influenced by the Madhyamika who are most likely to lay claim to the term ‘Middle Way’, and might even take offence at any attempt to separate it from Buddhist tradition, so an explanation is needed here of how the aims of the Middle Way Society differ from those of Mahayana Buddhism.
Perhaps the most important distinction between the Middle Way in Madhyamika and the universal Middle Way is that Madhyamika treats the Middle Way as a way of talking about an ultimate reality of some kind. This ontology is one of the ultimate emptiness of all phenomena, but nevertheless the concern of all the texts is with showing that metaphysical beliefs are ultimately unreal. This can be contrasted with the focus in universal Middle Way Philosophy on the Middle Way as method. Whilst the Madhyamika does have a practical motive, this practice is focused on realising the truth of emptiness by reflecting on the conditionality of all phenomena, rather than on making incremental practical judgements about all our experiences that use the Middle Way as a principle.
This is an important distinction, because the idea that the Middle Way offers an insight into reality can still be used as the basis of an appeal to authority. Madhyamika is made compatible with the authority of the Mahayana tradition and its gurus through the idea that its gurus have insights into the nature of emptiness as a sort of subtle reality. These gurus (for example, incarnate lamas or rimpoches in the Tibetan tradition) thus become idealised as individuals, the uneven positive qualities they often possess treated as a basis of authority discontinuous from ordinary experience, and the traditional dogmas of each Buddhist school are passed on through them. If the Middle Way was treated just as an incremental model, this kind of idealisation of teachers and tradition would not be possible, as all spiritual progress would be understood subject to asymmetries and as a matter of degree.
Nagarjuna dealt with the discontinuity between the ultimate emptiness of everything and our experience of things through his doctrine of two truths. This means, in effect, that we have to switch between an ordinary ‘truth’ and an enlightened ‘truth’ like a toggle switch, with the insights provided by ultimate emptiness somehow filtering into our ordinary experience. This is a very unsatisfactory way of thinking about the Middle Way, because it makes it irrelevant to ordinary experience. Even when you’ve supposedly experienced ultimate emptiness, this may or may not change your way of thinking about what you experience in everyday life. This problem arises from a mistaken way of thinking about the Middle Way in the first place. The Middle Way should not involve us in making even very subtle claims about a reality of any kind, but rather be a method we can apply at any level of experience from the grossest everyday life to the subtlest meditation. You don’t need an experience of ultimate emptiness to start applying the Middle Way, but just to recognise metaphysical assumptions and start changing your judgements.
The discontinuity in the Madhyamika account of the Middle Way also tends to go with an ego-killing view of our goals. If you think there are two discontinuous views of the world, one deluded by an ego (everyday awareness) and the other not (enlightened awareness), the only way forward is to get rid of the ego that is causing the delusion. In Middle Way Philosophy, on the other hand, the ego is seen as the positive basis of all progress. We just need to stretch and integrate that ego rather than kill it in order to make gradual spiritual progress.
It would be quite possible to practise the Middle Way in the universal sense whilst working within the tradition of Mahayana Buddhism and being inspired by the work of philosophers like Nagarjuna. However, as with any other tradition, to make it compatible with the Middle Way it is necessary to decisively avoid the metaphysical beliefs that go with appeals to authority in that tradition, and which tend to follow beliefs about a reality that has been reached by the leading authorities in that tradition. In some ways it might actually be harder to follow the Middle Way decisively when the very same term is constantly understood in an ontological way in all the texts and teachings that people use for inspiration in that tradition. However, Mahayana Buddhism in the West is also open to debate in a way that might raise some hopes that its interpretation of the Middle Way can be reformed.