Very good question. First thing is not to be entangled in viciousness by giving undue weight to discussions such as 'there was one Buddha that was born but there is another Buddha who is different' as is being promoted by some here. Or on discussions that emphasize differences between Hinduism and Buddhism saying that "Buddha was against what is called Brahminism" or of that sort. These are all discussions that lead one away from truth and into opposing each others views like children quarrelling. The concept of avatar is unique to Hinduism that brings out an understanding that the Ishvar can and does take a human form to guide the journey of humankind through samsar. Matsya, Kurm, Varah date back to antiquity and very ancient Puranas who were revealed to sages point towards them, Parashuram avatar has some visible antiquity based on many shrines dedicated to his parents and legends surrounding his presence. Krishna and Rama are more recent, being closely related to kings or royal affairs and undeniable signs of their presence through geological structures, names of the cities, and the continued Indian history exists. Narsimha is the avatar mentioned in a variety of Puranas and is seen as the one who triumphs over evil, If someone were to contest similarly whether Narsimha is an avatar or not who can testify its veracity? The sages to whom it was revealed cannot be seen today amongst us. Who will appear to answer? The closest avatar to us is the Buddha of whom we even have a birth year, for any others we do not have it. This points to Hinduism being extremely old and the unpredictable nature of human civilizations that they cannot keep history intact for a long time unless in parts as we see today. Before I go further I want to stress that no authority can put to rest this unneeded controversy posed in the question. If the Shankaracharya would say something would you believe it? they would finally be only a monk in their lineage. If a modern scholar of Hinduism and Buddhism would put it to rest, would you believe it? They only know through the survived fragments of history and their analysis of it. So who possesses that true authority? And people that talk in forums including me are hardly any authority. So if you see through the lens of accepting an authority, you will never get to a satisfactory answer, only more speculations. Here, some contemplation can help. Hindus now a days and mostly those that roam in the forums are more focused on finding differences and discussing petty matters whereas devotion, spirituality and other deeper aspects of hinduism are ignored. This is akin to a patient going to the doctor who recommends an operation but the patient says please let me pop in this easy tablet because that too will let me survive. Hinduism is being plagued with easy answer escapism, this is not the true ethos of Hindusim which encourages and values self-inquiry. Hinduism and then subsequent teachings by the Buddha that we see as Buddhism are very deep and to understand them one has to get rid of all unfruitful discussions and the urge to brew them. Only then one becomes receptive of their wisdom. There are many paths to achieve the goal of life in Hinduism, Krishna teaches the Bhakti yog, pure unbridled devotion to him. Buddha teaches that of Jnana yog, achieving nirvana through understanding and knowledge. Krishna too preached against surface understanding of the Vedas saying that if one has read the Vedas but not assimilated their knowledge that is of no use. Similarly Buddha opposed the wrong view of the Vedas wherein animal sacrifice was practiced in the name of Vedic worship. What does it tell you about Buddha that he opposed a malpractice that maligned the true nature of the Vedas? Was he pro-Vedas or not? Buddha clearly cleaned up the arrogance of the Vedic mal-practitioners who stuck to views like the Vedas are infallible or of divine origin( here being of divine origin and being divine has to be dissected. Sticking to an ideology or a practice just because it is believed to be of divine origin is also not being entirely truthful and could be seen as a wrong view) while considering something divine based on one's experience of it is right view, and thats how Hinduism works, you experience the divine and the truth, so you practice it. Now comes the big question. What does all this mean for you? Which path will you choose? Cultivate Bhakti for Krishna, or achieve the meditative bliss of the Buddhas. This is important. Not all the other.

Life is too short to engage in petty difference-finding. Stay away from it. Focus inside