I think most people would agree that Buddhist doctrines first originated within a Hinduism-informed general milieu, and they can only be understood comprehensively in this light. Whereas Jainism seems to have developed in parallel with Vedic religion, and to have shared some of the same underlying concepts. Whether this means either are "part" of Hinduism probably depends on whom you ask.

There are thousands of sects of Hinduism and most still list themselves as such until the British came in

The word Hindu itself is an exonym. What you are doing is playing language games by conflating everything Indian with Hindu.