You're basically on point with that. https://investor.vanguard.com/investment-products/mutual-funds/profile/vtsax https://investor.vanguard.com/investment-products/etfs/profile/vti In fact looking at it VTI is cheaper by expense ratio. Really though if you're looking for retirement stuff, you can't really go wrong with doing mutual funds and hitting something like a retirement target age thinger like VFF(whatever your retirement year is) VFFVX is for 2055. Those take riskier stocks early on to make money and as you get closer sell off stocks and buy into more stable stuff like bonds to ensure you're getting your full retirement money without magically losing it all in the end.
Although in general what most of those retirement funds do is just invest in tracking the stock market (S&P 500) since it is ON AVERAGE returning 7% a year. Which means you double roughly every ten years. You start with 10k that you never add to in your whole life at 20 years old? 20k at 30yo, 40k at 40yo, 80k at 50yo and 160k at 60yo. Hence the power of investing early.
But really a mutual fund if it's supposed to be for retirement is ACTIVELY managed, but the reality is they just stick with the good old sandyp and reap the benefits of being a real person "managing" a fund. Now other mutual funds that aren't aimed for retirement I can't say much about them negative or positive never interacted with them.
An index fund is basically the same as a mutual fund, but again it's just a robot or formula to mimic the sandyp 500 and get that 7% a year return. Except because there's no real person "managing" it you save money on expense fees.
ETFs are ALMOST an index fund, except they're just an index fund that gets actively traded like stocks how you mentioned. Index funds are more accessible via your job benefits similar to mutual funds. Outside of that you'd have to finagle more to jump on mutual or index funds that aren't just tracking the sandyp 500.
There are ETFs that do a variety of things such as tracking large cap companies etc. Index funds do the same and mutual funds as well.
TL;DR: Mutual funds have real people managing which stocks they buy and you gamble on a real person actively looking for the best deals to make the most money (but if it's for retirement the best option in general is to follow S&P 500). Index funds and ETFs just follow an algorithm like tracking the S&P 500 and auto buy those stocks to follow that graph or whatever graph the formula tells them to (could be follow the trend of ESG bond companies for example super duper safe fund/stock).