Nancy, thank you for your good and important question. I will try to take a stab at some points and hope that some other folks will pick up where I left off and provide more insight and more details.
There seems to be some problems, both of reality and perception, with the quote from Huston Smith, and I think that your hunches are going in the right direction.
1. This is not a value judgment one way or the other, but I find the whole idea of the influence of Buddhism fairly questionable. That doesn't mean that there couldn't have been or that there presently are Hindus (maybe even prominent ones--e.g., Mahatma Gandhi or Ramakrishna), but as far as the religions go per se, the influence seems to have gone pretty much in the other direction. After all, Buddhism arose within the context of Vedantic Hinduism, within which emerged the Upanishads and the philosophical schools. Theoretically, Buddhism could have been just one more of these different schools.
2. However, from its very outset, the teachings of the Buddha were considered to be out of sync with mainstream Hinduism. For one thing, it rejected the authority of the Vedas, and, consequently, Buddhism occupied an outsider's niche within the Hindu culture.
3. For another thing, Buddhism rejected the caste system, which was beginning to reach its full stride just alongside Buddhism. It's hard to make a case for Buddhism's effectiveness against the caste system over all of that time. It is true, however, that presently Dalits (untouchables)areturning to Buddhism in order to escape their status. Still, the (just a little more successful) efforts to abolish the caste system have been catalyzed from the outside and only then became a goal for a number of Hindus as well.
4. The principle of ahimsa, which means the prohibition of harm ing any living being, has always been a doctrine of Buddhism and Jainism and has never been an integral part of Hinduism. Again, some prominent persons, such as Gandhi may have picked it up from those other religions. One easy example that ahimsa is not a Hindu principle is the second chapter of the Bhagavad Gita, in which Krishna instructs the warrior Arjuna to fulfill his caste duty and participate in a battle against his cousins' army. Furthermore, Krishna says, there's really no problem in killing people since doing so only harms a person's body, but not their immortal soul.
Thus, I really can't wrap my mind around the statement by Smith you cite. I'll leave it with that for the time and hope we'll get some more input from other interested readers. Karl-Heinz.
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.