What is the point/story of origin of the Sanatan Dharma and why has it been altered as Hindu Dharma?
Why "Hindu Dharma" and "Sanatan Dharma" are synonym? :
First of all, the Sanskrit term Sanatan means eternal or ancient.
Sanātana Dharma (Devanagari: सनातन धर्म meaning "eternal dharma" or "eternal order") has been proposed as an alternative, "native" name for Hinduism (Hindi Hindu Dharm हिन्दू धर्म "Hindu religion". The term was mentioned and explained in depth in Vedic literature (Rig Veda) (4-138) and was used during the Hindu revivalism movement in order to avoid having to use the term "Hindu" which is of non-native (Persian) origin.
According to सनातन धर्म:
वैदिक काल में भारतीय उपमहाद्वीप के धर्म के लिये 'सनातन धर्म' नाम मिलता है। 'सनातन' का अर्थ है - शाश्वत या 'हमेशा बना रहने वाला', अर्थात् जिसका न आदि है न अन्त।
English: In vedic era, the name "Sanatan Dharam" is founded for the Dharma of Indian sub-continent, The meaning of "Sanatan" is eternal, 'Would remain forever' i.e. which has no beginning and ending.
सनातनमेनमहुरुताद्या स्यात पुनण्रव्
Meaning: Sanatan is called which is innovative/renewed even today.
"यह पथ सनातन है। समस्त देवता और मनुष्य इसी मार्ग से पैदा हुए हैं तथा प्रगति की है। हे मनुष्यों आप अपने उत्पन्न होने की आधाररूपा अपनी माता को विनष्ट न करें।"
Meaning: This path/route is sanatan. All deities and human beings are born and progressed from this....
भारत का सर्वप्रमुख धर्म हिन्दू धर्म है, जिसे इसकी प्राचीनता एवं विशालता के कारण 'सनातन धर्म' भी कहा जाता है
English: It is called "Sanatan Dharam" for its ancientness and hugeness
Important note from Britannica Encyclopedia:
Sanatana dharma, in Hinduism, term used to denote the “eternal” or absolute set of duties or religiously ordained practices incumbent upon all Hindus, regardless of class, caste, or sect. Different texts give different lists of the duties, but in general sanatana dharma consists of virtues such as honesty, refraining from injuring living beings, purity, goodwill, mercy, patience, forbearance, self-restraint, generosity, and asceticism. Sanatana dharma is contrasted with svadharma, one’s “own duty” or the particular duties enjoined upon an individual according to his or her class or caste and stage of life. The potential for conflict between the two types of dharma (e.g., between the particular duties of a warrior and the general injunction to practice non-injury) is addressed in Hindu texts such as the Bhagavad Gītā, where it is said that in such casessvadharma must prevail.
The term has also more recently been used by Hindu leaders, reformers, and nationalists to refer to Hinduism as a unified world religion. Sanatana dharma has thus become a synonym for the “eternal” truth and teachings of Hinduism, the latter conceived of as not only transcendent of history and unchanging but also as indivisible and ultimately nonsectarian.
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The term sanatana has very clearly been defined in the above answer. It literally means "eternal". Dharma means "occupation".
Sanatana Dharma hence is translated as "Eternal Occupation".
Before concluding the question, I consider it very important for people to understand what religion actually means, by logic and science. Without this understanding one will be dwelling either to fanaticism or mental speculation. Hence, shall never be able to understand the seemingly mythical texts which in actuality is a subset of Absolute Truth.
This has very nicely been explained by the world's most widely published and accurately translated Bhagavada Gita, in its authentically verified introduction.
The English word religion is a little different from sanātana-dharma. Religion conveys the idea of faith, and faith may change. One may have faith in a particular process, and he may change this faith and adopt another, but sanātana-dharma refers to that activity which cannot be changed.
For instance, liquidity cannot be taken from water, nor can heat be taken from ﬁre. Similarly, the eternal function of the eternal living entity cannot be taken from the living entity.
Sanātana-dharma is eternally integral with the living entity. When we speak of sanātana-dharma, therefore, we must take it for granted on the authority of Śrīpāda Rāmānujācārya that it has neither beginning nor end.
That which has neither end nor beginning must not be sectarian, for it cannot be limited by any boundaries. Those belonging to some sectarian faith will wrongly consider that sanātana-dharma is also sectarian, but if we go deeply into the matter and consider it in the light of modern science, it is possible for us to see that sanātana-dharma is the business of all the people of the world – nay, of all the living entities of the universe.
Please understand 2+2=4 universally. So is religion, it is valid universally. The laws of karma, the heat & light of fire, the liquidity of fire are universal concepts. They are religion, eternal, unchanged.
Similarly, religion begins when we understand that we are also part of religion, which is, we are spirit souls, not the dying and changing body. That is why Bhagavada Gita begins with knowledge of indestructible, immortal, eternal, spirit soul. (that's us, encased within the body).
It is very clear now that religion means that which has no beginning, no end, it has no boundaries. Hence religion cannot be restricted by a specific label such as Hinduism, Judaism, Islamism, Humanitariansism, Vegeterianism, Philanathropism and any -isms.
These are labels to help people attain self-realization as per their comforts but the terms are not Absolute Truth. We will read ahead, throughout the Vedas, there is no term such as "Hindu".
A person born to a lawyer cannot be said to be a lawyer. Similarly, a person cannot be judged as Hindu if born in Hindu. Same for any other followings.
Just as a gold found in America or India, is a gold, it is not American gold or Indian gold, simply gold, same chemically. Similarly, every soul is equal and beyond such designations.
Why "Hidnu Dharma" and "Sanatan Dharma" are synonym?
It has been found being quoted by many great well known philosophers and swamis. Even in diaries of many recent saints. One such quotation is shared herewith from today's most authentic establisher of lost Vedic wisdoms, A.C Bhaktivedanta Swami Srila Prabhupada:
This word Hindu is not a Sanskrit word. It was given by the Muhammadans (Persians). You know that there is a river, Indus, which in Sanskrit is called Sindhu. The Muhammadans pronounce s as h. Instead of Sindhu, they made it Hindu. So Hindu is a term that is not found in the Sanskrit dictionary, but it has come into use. But the real cultural institution is called varnashrama. There are four varnas (social divisions)--brahmana, kshatriya, vaishya, and shudra--and four ashramas (spiritual divisions)--brahmacharya, grihastha, vanaprastha, and sannyasa.
Therefore, in substantial, at it's naked truth, the term Hindu has recently been coined. Before that, throughout Vedic ages, there was no such terms, it was a common platform where everyone loved and obeyed instructions of the Lord for that is what religion means as confirmed throughout our Vedas. Confirming the same:
sa vai puṁsāṁ paro dharmo yato bhaktir adhokṣaje
ahaituky apratihatā yayātmā suprasīdati
The supreme occupation [dharma] for all humanity is that by which men can attain to loving devotional service unto the transcendental Lord. Such devotional service must be unmotivated and uninterrupted to completely satisfy the self. (SB 1.2.6)
In order to trace the origins of Sanatana Dharma and hence Hindu Civilization, the evidence that is available must be examined from multiple perspectives and must corroborate for any dating to be certain. To date, various methods have been used to assign dates--but they do not concur on many aspects and thus the process of dating Sanatana Dharma is one that is ongoing.
Gandhi ji never attempted to give a codified form of to the Hindu Faith. In fact he recognized that Hinduism developed into a queer but ever evolving mixture of beliefs , practices and traditions. He also recognized in the process of evolution many excrescences like untouchability, child-marriage, Sati, animal sacrifices to gods, etc, had during the passage of time , clung to sanatana Hinduism and the faith had to rid of all of them if it was to survive and thrive.