Sure, index funds can track any sort of index, and use any sort of weighting scheme. But modern portfolio theory dictates that truly passive investors should hold two assets: the "market portfolio", and a "risk-free" asset such as Treasury bonds, or cash.
As other commenters have mentioned, "index investing" is better referred to as "passive investing".
I think the very term "indexed fund" is an artifact. Indexed funds need not be tied to an index. They can still be "passive investing" without the index.
"Antitrust" is a similar linguistic artifact as you can violate antitrust rules without any mention of trusts,
I agree we should get away from the term "index fund", but for better or worse, investors are hung up on indices. Why do we talk about the S&P 500? Why not 1k? etc.
Ick, a third overlapping term. I would call purely quant funds a form of indexed funds in that they follow a specific set of rules. Those rules are whatever procedure, computer program, or other non-human algorithm is used to pick investments.
Whether or not a quant fund is passive investment is semantics imho. I wouldn't call spending hours/days/weeks examining a quant funds system a passive activity. And investors can move in and out of funds quickly these days. On the other hand many see passivity wherever investors or fund managers put buy/sell decisions in the hands of non-humans.
As other commenters have mentioned, "index investing" is better referred to as "passive investing".