And if you went with that maximalist definition of an index, then the word would have no useful meaning. If the inclusion criteria is completely arbitrary and capricious (i.e. whatever cathie woods decides she likes that week), then it's still a fund, it's just not an index fund.

QQQ is both arbitrary (the stock exchange has nothing to do with business operations) and capricious (they are changing the rules on a whim to serve Elon's interests, and because the arbitrary inclusion criteria somewhat forces them to).