But consider the question: "What is Buddhism, as defined by prominent Buddhists themselves, whose prominence is recognized by traditional institutions that uphold Buddhism?" It doesn't seem to me the answer to this question would have changed much during the last millennium.

This chap would disagree. There's rather a lot of words there, so briefly: Buddhism in the Western world -- what he calls "Consensus Buddhism" -- is for the most part an invention of the 19th and 20th centuries with more roots in European and American culture than in the countries it came from.