There is no real-Buddhism but rather a gigantic tradition, a great Dharma Mountain with many meandering footpaths, refuges, and activities on its slopes. This ‘Great Dharma Mountain’ includes everything and also its opposite and yet remains one single Mountain. In fact, there were 84,000 paths taught by the Buddha 2,500 years ago and many more discovered since – many paths, one Mountain.
|Celestial Buddha, Amitabul, in the Pure Land Heaven|
Almost every Buddhist will tell you what Buddhism is ‘really‘ but this tells more about the likes and dislikes of the author than Buddhism. Buddhists will often insist on which beliefs and practices should be accepted and which ones disguarded and not surprisingly their favourite bits are the parts they emphasise while the others aspects, they maintain, should be discarded (how fortunate it’s not the other way round). The truth, however, is that there is a huge variety of practices and beliefs that are included in Buddhism just like there are many ways to skin a cat or many paths to climb a mountain.
There are certain strands that one can identify within the tradition but none are quintessentially Buddhist. Even the famous ‘no-self’ (anattā) which despite being a constant strand and sometimes said to be the defining feature of Buddhism it has still caused heated debate down the centuries and some sutras such as the Nirvana sutra go as far as saying:
“in truth there is the Self [ātman] in all dharmas
[phenomena]” and “The Self (ātman) is reality (tattva), the Self
is permanent (nitya), …the Self is eternal (śāśvatā), the Self
is stable (dhruva), the Self is peace (siva).”
Furthermore, other parts of the tradition argue that ‘no-self’ is to be understood as a sceptical teaching where one frees oneself from attachment to all metaphysical views and both ‘self’ and ‘no-self’ are views one must overcome.
Within the Buddhist tradition you find every kind of idea presented and also its opposite. As already mentioned you find diverse ideas such as ‘no-self’, its opposite ‘eternal-self’ and its counterpart ‘no-views’. You find atomic realism (eg, Sarvastivada) and all kind of opposites such all reality being the creation of Mind (eg, Yogacara) or an illusion (eg, Avatamsaka) and many other teachings besides.
Some believe that Buddhism denies gods (eg, much of the Western school) while others
pray daily to celestial Buddha’s and Bodhisattvas (eg, Pure Land Buddhists) and even talk of God (eg, some Zennists) . Some strains will teach that Buddhism has been revealed by Buddha and passed on in its almost perfect form (eg, Southern Buddhists) while others think that the teachings are not to be taken literally but discovered for oneself (eg, Zen). Others insist Buddha was a political liberal who was interested in reforming society while rivals claim he renounced society and politics altogether. In the Mahayana ‘emptiness’ (sunyata) is sometimes claimed to be fundamental but that would mean that nobody understood ‘real’ Buddhism until hundreds of years after the death of the Buddha including the Buddha’s own disciples.
In fact, the Venerable Ananda said in the Pali-cannon that there were 84,000 teachings he knew of (Theragatha 17.3 (vv. 1024-29)) which indicates the shear magnitude of teachings given while the historical Buddha was still alive which have magnified beyond measure through the teachings of enlightened masters since then.
It is safe therefore to conclude then that when anyone tells you that Buddhism is X,Y and Z or Buddha was ‘like this’ or ‘like that’ then they are talking about a single strand (or several strands) of the Buddhist tradition and are excluding the rest based on their own preferences. No presentation of Buddhism or the Buddha can possibly encapsulate the whole variety of ideas, stories and practices. Alarm bells should ring whenever anyone claims that they have discovered the correct strand of this huge tradition whilst the rest is an unnecessary add-on or mistake. Claims of this type are reflections of the mental state of the author and tells you little about what Buddhism is actually like. There is not one ‘true Buddhist tradition’ any-more than there is one ‘true-tree’ on a mountain. To deny the other beliefs and practices is committing the ‘no-true-Scotsman fallacy’.
The only strands that run undisputed through the whole tradition is a certain ethical code
|Buddha in meditation|
and the practice of mediation but even here different schools highlight some methods and practices and downplay or deny others. And even in mediation there are different understandings about the same practices leading to endless confusion and debate. There is not even agreement about the goal of meditation and different schools, teachers and sutras give a variety of ideas and interpretations about ‘enlightenment’ such as overcoming life and death (eg, Southern school), realising Buddhahood (eg, Eastern schools) or seeing the world as it really is (eg, many Western Buddhists).
To conclude then: there is no one ‘true-Buddhism’ to be separated from a ‘wrong-way’ but rather a Great Dharma Mountain of endless teachings. So let us enjoy the journey up the Mountain and I’ll race you to the top by whatever way you choose to travel.
I can't help but feel that there must be some common thread that connects the various Traditions.
To use the analogy of weaving:
If there is a Warp [The Four Seals]that the threads [The Various Buddhist Traditions] are bound together with would allow some common basis for ascribing the label Buddhist to the disparate strands.
DZONGSAR KHYENTSE RINPOCHE states that:
"One is a Buddhist if he or she accepts the following four truths:
All compounded things are impermanent.
All emotions are pain.
All things have no inherent existence.
Nirvana is beyond concepts.
These four statements, spoken by the Buddha himself, are known as “the four seals.”
Does anyone more knowledgeable than myself know of any Buddhist Traditions that do not ascribe to the above Four Seals?
Thank you for answering and asking questions. Anyway, I think the dharma-seals (法印) despite sometimes being supposed to be the mark of being Buddhist nevertheless don’t qualify for a quintessential account of Buddhism (wishful thinking, I think, from some Mahayana authors).
Our first clue here comes about because there is no agreement about their number or content. There are several versions and they have been offered as one, three, four or five seals depending on the tradition (perhaps there's others that I have not heard about).
VARIETIES OF DHARMA SEAL [法印]
1) The one seal [一法印] is a Mahayana concept developed in opposition to what some thought was a Hinayana concept. The one seal represented ultimate reality, or the true aspect of all phenomena based on the teachings of the Lotus Sutra.
3) The three seals [三法印] are said to be impermanence, non-self, and nirvana.
4) The four seals [四法印] are impermanence, suffering, no-self, and Nirvana.
5) The five seals [五法印] are impermanence, suffering, no-self, Nirvana and emptiness.
DHARMA SEALS AND INHERENT EXISTENCE
Seals one and three of your list are not particularly Buddhist and Christians, Muslims, Nihilists, and others would all agree. The third seal you quote as “All things have no inherent existence” but then what about the claim of Buddha in the Pali-cannon that Nirvana is unconditioned? In the Pali-cannon Buddha is recorded as saying (Saṃyutta-nikāya I),
"O bhikkhus, what is the Absolute (Asaṃkhata, Unconditioned)? It is, O bhikkhus, the extinction of desire (rāgakkhayo) the extinction of hatred (dosakkhayo), the extinction of illusion (mohakkhayo). This, O bhikkhus, is called the Absolute."
Thus is it understood in Southern Buddhism that nirvana is an uncompounded or unconditioned state of being which is "transmundane". The Mahayana also have candidates for inherent existence such as the tathagatagarbha (如来藏), buddhadhatu, buddha-nature (佛性), Mind (vijñapti-mātra), and so on.
It has caused endless debate how things can have no-self, be empty, and have no inherent existence when the tathagatagarbha is said to have an element that is "essential, immutable, changeless and still". The tathagatagarbha is apparently equated with the "original edge of reality" (bhutakoti) that is “beyond all distinctions, the equivalent of original enlightenment, or the essence” (Vajrasamadhi Sutra).
ANSWER (PART TWO)
In modern times we still have disagreement about this issue and we find scholar-monks such as Thanissaro Bhikkhu  saying:
“The idea of emptiness as lack of inherent existence has very little to do with what the Buddha himself said about emptiness. His teachings on emptiness — as reported in the earliest Buddhist texts, the Pali Canon — deal directly with actions and their results, with issues of pleasure and pain. To understand and experience emptiness in line with these teachings requires not philosophical sophistication, but a personal integrity willing to admit the actual motivations behind your actions and the actual benefits and harm they cause.”
In this respect in the past decades in Japan there has been on ongoing debate on whether Zen and the Mahayana should qualify as Buddhist at all  and their basic argument is around Buddha nature, inherent existence and related issues are a deviation from the real Buddhist teachings anyway.
Incidently, scholars such as Harvey and Kwoem say the only aspect that can really be said to make someone Buddhist is refuge in the Triple Jewel (Buddha, Dharma and Sangha) as doctrine varies too much among schools and traditions. This is relevant here, I think, because if it were really the case that the seals (法印) offered a solution to what Buddhism was then Buddhist scholars would not have overlooked it but rather the claims come from popular teachers with a vested interest in simple and neat answers (Buddhist teachers are the salesmen and sales women of the Dharma-world, so to speak).
So, I would say that while there’s certainly something in the idea of the seals (法印) and it captures some mainstays of the tradition it is too much to say this is what would make something Buddhist or not.
 Swanson, Paul L. (1997), "Zen is not Buddhism, Recent Japanese Critiques of Buddha-Nature", Numen, Brill Academic, 40 (2),
 The Integrity of Emptiness" Access to Insight, 10 January 2017, http://www.accesstoinsight.org/lib/authors/thanissaro/integrityofemptiness.html