Two Buddhas in Hinduism and Buddhism

Historically there were two Buddhas who took birth in ancient India, but historians and religious hindu gurus tried to project both of them as same person as they felt the purpose of both was the same.
two buddhas in history

The first Avatara Buddha, who was incarnation of Lord Vishnu, established the philosophy of Ahimsa(nonviolence) and convinced those followers of Vedic customs, who by that time had become bent toward animal sacrifice, to give up such rituals and become kind to animals.
He did not establish any godless or monist philosophy like the latter one.
This Avatara Buddha was born to his mother Anjana in a place, what became known as Bodhgaya in later years.
He is the actual avatara of Vishnu.
Infact, he was mentioned even by asura king Ravana many years ago.

The second Buddha known as Gautama or Siddhartha was born as a prince in Lumbini, now in Nepal, to Mayadevi in 1887 BCE.
He is the one buddhists refer about, the prince who left home to do austerities to find enlightenment.
He went to Bodhgaya to meditate because of its spiritual potency as the birthplace of the avatara Buddha.
gautama buddha or siddhartha
Then he became enlightened to the reasons for suffering in this world, and developed a godless way of becoming free from suffering.
Since then, he established the monist and godless philosophy of Buddhism, which became named after him by his followers.

The word Buddha in Sanskrit means “The Awakened One“, one who is awakened to Reality, who understands true nature of the mind, the world, and all sentient beings.
The Theravadin texts refer to six preceding Buddhas (those who have awakened) as Vipasyin, Sikin, Krakuccanda, Konagamara, Kashyapa, and Maitreya as the Buddha of the future.
Everyone who is awakened his Buddhi(sense) can become a Buddha.
Bodhi” means “Awake”, and “Sattva” means “sentient”. The term “Bodhisattva” refers to the awaked ones who has sympathy for the humanity absolutely as well as show empathy for them and be ready to help them. Generally, people often use the word “Bodhisattva soul” for those who are good-hearted.
In order to become the Buddha, the Buddhist must go on the Bodhisattva path.
Buddha and Bodhisattva

But here, we are talking about the Vishnu’s avatar Buddha and Gautama Buddha, whose teachings are followed by Buddhists worldwide at present. Infact, Gautama Buddha himself did not create a religion. His followers went to other kings, mentioned that Gautama was born as Siddhartha in Ikshvaku dynasty in which Lord Rama also was born 55 generations ahead. So, he is incarnation of Vishnu and we all should follow him. Biggest irony at that time was that, while Gautama Buddha wanted everyone to equally practice his principles irrespective of caste, his followers convinced Kshatriyas (Kings) that Siddhartha was born in their caste, so they should take pride in that matter and follow him.

Adi Sankaracharya, while discussing or arguing with others, did not discriminate between the purpose of one or the other.
Sankaracharya, who was born more than 1300 years after Gautama Buddha, developed his own sunya philosophy, which was very much like the Buddhist philosophy, replacing the Buddhist nirvana with his Vedic Brahman, to defeat Buddhism and drive it out of India.
He succeeded most effectively, as at that time many were leaving Vedic culture and converting to Buddhism.
But with this new Mayavadha philosophy from Sankaracharya, Buddhism bowed, conversions stopped, and Buddhism itself started to decline.
This is one main reason why these two Buddhas became merged into one identity.
Later many Acharyas in India defeated Sankaracharya’s monist or impersonalist Mayavada philosophy but the purpose of stopping conversions of hindus into buddhism was already achieved.

Srila Jayadeva Goswami’s composition “Gita Govinda” describes 10 avatars of Vishnu as :

vedan uddharate jaganti vahate bhugolam udbibhrate
daityam darayate balim chalayate kshatra kshayam kurvate
paulastyam jayate halam kalayate karunyam atanvate
mlecchan murccayate dasaktikrite krishnaya tubhyam namaha

Translation : Krishna, He who took ten incarnations! I offer my obeisances unto You for saving the Vedic scriptures as Matsya-incarnation; You help up the universe as Kurma-incarnation, and lifted up the world as Varaha, the Boar incarnation; as Nrishimha You vanquished Hiranyakashipu; as Vamana You deceived Bali Maharaja; as Parasurama You exterminated the corrupt warrior class; as Rama You slew Ravana; as Balarama You took up the plough; as Buddha You bestowed compassion, and as Kalki You kill the Mlecchas.

In his Dasa Avatara Stotram, Srila Jayadeva writes in the ninth verse:

nindasi yajna vidherahaha shrutijatam
sadaya hridaya darshita pashughatam
kesava dhrita bhuddha sarira
jaya jagadisha hare jaya jagadisha hare

Translation : O Lord of the universe, Kesava! You took the form of Lord Buddha Who is full of compassion and stopped the slaughter of animals which is strictly forbidden in the Vedas.

The above Buddha is clearly the incarnation of Lord Vishnu.

Infact, puranas do not mention concept of Dasavataras. Vishnu had many other avataras like Hayagreeva, Mohini, Venkateswara which were not included in that list of 10.
Books like Amara Kosa, Prajna-Paramita Sutra, Astasahastrika Prajna-Paramita Sutra, Sata-shastrika Prajna-Paramita Sutra, Lalita Vistara, etc describe multiple Buddhas.
Above books reveal the existence of three categories of Buddha, namely:
Human Buddhas: like Gautama, who came to be known as Buddha after enlightenment.
Bodhisattva Buddhas: Personalities like Samanta Bhadraka, who were born enlightened.
Adi (original) Buddha: the omnipresent Vishnu’s Avatara of Lord Buddha.

Adi Buddha
Lankavatara Sutra is a famous and authoritative Buddhist scripture.
From the description of the Buddha in this book, it can be concluded that he is not the more recent Sakya Simha or Gautama Buddha.
In the beginning of this book we find Ravana, King of Lanka, praying first to the original Vishnu incarnation Buddha and then to the successive future Buddha. A part of this prayer is :

lankavatara sutram vai purva buddha anuvarnitam
smarami purvakaih buddhair jina-putra puraskritaihi
sutram etan nigadyante bhagavan api bhashatam
bhavishyatyanagate kale buddha buddha-sutas cha ye

Translation : Ravana, the king of Lanka, at first recited in the Totaka metre, then sang the following – ‘I invoke in my memory the aphorisms known as Lankavatara-sutra, compiled and propagated by the previous Buddha (Vishnu’s incarnation). The son of Jina (Lord Buddha) presented this book. Lord Buddha and his sons, who will appear in the future, as well as Bhagavan, the Vishnu incarnation, will continue to instruct all from this book.

Lankavatara Sutra is wrongly depicted as conversation between Gautama Buddha and a Bodhisattva named Mahamati.
Infact, it is conversation between Aadi Buddha (original Buddha) and his follower Mahamati. It is set in Lanka, which Ravana ruled later.

This proves that the first original avatar of Vishnu came on earth long long before Gautama Buddha and Ravana, who lived in Treta Yuga knew about him.
As the cycle of four yugas keep repeating, Ravana, in some treta yuga must have worshipped Buddha (who came much earlier in Krita Yuga) from one of his previous Kali or some other Yuga.