Indians claim everything to be their invention. Even flight and space travel, where apparently they have perfected it millennia ago. Their tales are as credible as the cow dung they use in their everyday life.
Buddhism didn't originate from "India". It originated from a region that lies between modern day Nepal and India. If anything, Nepal has a stronger claim than India because the Buddha was actually born there. When Xuanzang arrived at the Indian subcontinent, he never mentioned about any nation called "India". Instead, he mentioned the names of numerous kingdoms like Lampa, Kashmira, Mathura, etc. But in all the English language translations of his Journey to the West, they mentioned that he traveled to "India". For modern China, I think it's for convenience sake. But for the rest of us, it's the Anglos and the Indians trying to reinterpret history. We should actually be reading the original documentations of Xuanzang and other documents that were originally from that era. We cannot fully trust Anglo documents.