The influence Buddha had on Hinduism was more reactionary. The challenge Buddhist philosophy bought to Hindu philosophy lead to significant movements within Hinduism to answer those challenges, such as the Advaita movement founded by Adisankarcharya. Advaita has been accused of being crypto-Buddhist, because it seems to incorporate some of the philosophical elements within Buddhism from a Hindu point of view such as the doctrine of no self. In actuality, this accusation is exaggerated, because justification can be found for Advaita within the Upanishads itself, which is older than Buddhism. However, the impetus to formulate the Advaita philosophy seems to have resulted from the conflict amongst the different schools of philosophy in India on core issues(such as self vs no self, being vs no being, reality vs illusion)
The inclusion of Buddha amongst the avatars of Vishnu list is again reactionary in order to appropriate the Buddha of Buddhism. The justification for this is that Buddha was not a challenger of Hinduism, but a reformer. It was his followers who formed the religion of Buddhism based on misinterpreting what he said.