This is not a direct answer to the question, but will perhaps round things out.
I once hung out with some devout, practicing (Asian-descent) college-age Buddhists who had taken college courses on Buddhism, and the topic of the similarity between Buddhism and Hinduism came up. At least in this particular conversation, these individuals agreed there wasn't any difference between Buddhists and Hinduism. I don't know, so I can't say what their reasons were for why they believe there is no difference. I imagine they were thinking of the portions of Hinduism that also reject the caste system and are open to non-Indian converts. Given the points above, they'd have to! But from what I know of these individuals, they're big on Buddhism's philosophical systems. So I assume they were concluding this on the basis of comparing philosophies. But maybe I'm doing them a disservice and they also meant to include practices too - e.g. cultic practices, meditation, temple worship, etc.
From my own knowledge of Hinduism, I think given how some Hindu treatises are devoted to disproving some Buddhist arguments, there has to be some influence.