I agree that the designation is largely orthogonal.
- Mutual funds and ETFs are two types of funds defined by how they trade. ETFs trade during the day in discrete shares like individual stocks, while mutual funds sequence the transactions after hours by the penny almost like a bank account.
- Both mutual funds and ETFs can also be subdivided based on what they track. Index funds track a passive index (like the S&P 500, Russell 2000, etc). This is in contrast to actively managed funds that are subject to the trading decisions of individual fund managers.
So mutual funds can be either active funds with a manager making the calls or passive index funds that follow a set index. And ETFs have historically been index funds, but new actively managed ETFs are also becoming trendy.
When in doubt, I personally prefer ETFs that track an index. They'll be low cost, available at virtually any brokerage, not subject to the whims of manager turnover, and (because of how they're structured) will rarely report any forced capital gains that will affect your tax bill.