I don't know about the funds, but it's really about the index. Both for the index funds that use the index, and the active mutual funds and index funds benchmark to that index.

Why is it really about the index though, if the index fund doesn't track that public index?

If the index fund is tracking some proxy that is float weighted, isn't that what matters? At least when it comes to people's money.

Index funds track an index, thus the name