There has been some talk here about reincarnation being introduced into Hinduism BT Buddhism. however ado not the Upanishads and Bhagavad Gita pre date Buddhism? These texts specifically teach the concept of an atman reincarnating. The Buddhist view of course is very different.
Most of the Upanisads (possibly a couple of exceptions) and the Gita are post-Buddhist, despite their own claims to be incredibly ancient.
I thought the Upanishads were the Vedic response to Buddhism. Is this right?
I am not sure. I haven't heard this view before, but that isn't saying much!
Some of them could be, but that would more likely be the later ones.
Because the early ones are around the start of Buddhism itself,
when Buddhism was still rather a minor influence in the bigger picture.
There are a lot of upanisads, so it may be difficult to generalize.