An Emergency Fund is designed to protect your assets, so that in the event of an emergency you do not need to liquidate assets at a price dictated by the market.
I personally disagree with this definition, and I think mpbaker22 does as well, which is why there's disagreement here. We're starting from different perspectives. I view an Emergency Fund as money I can access to cover emergency expenses. Full stop. Whether I have to sell stock a less than I bought it for is irrelevant. As long as the money is available for use in an emergency, it's an emergency fund.
I think it's much more important to have an actual savings account emergency fund when you're first starting out, but after you've accumulated significant assets, it seems inefficient to me to keep so much money sitting on the sidelines when you have plenty of other money you can access. The upside is greater than the downside, especially because I think the chance of having an actual emergency is pretty rare. Unemployment insurance, homeowners/renters insurance, car insurance, and health insurance all keep the risk of needing lots of money for an extended period to be pretty darn small.
I think you should reconsider the nature of the EF, even if you have to give it a special name in order to not change your notions of what an EF is, consider it from my perspective when calculating opportunity costs.
Now, I completely disagree about it being fine to sell for less than you would like because you are forced to do so by poor planning, that isn't good at all.
I do like your point about the insurances kicking in, but that then makes the EF easier to calculate, you just look at all the scenarios that could happen, and say it is a disability, and your premium starts paying after X days, you need the money to tide you over til the X date.
As for who needs one, I look at it as being akin to Insurance in that:
Lower Class (poor) don't need it, they have nothing to protect and should be putting all their money to paying off debt and building a nest egg.
Middle Class (upper/lower) do need it, they have built assets over time and they need to protect them.
Upper Class (HNW) doesn't really matter, they could choose to have one, or not, and if their equities tank it is fine up to a point.
The people who have achieved the FIRE Nut might well need one since the impact of doubling your withdrawal rate could be devastating, or if they have enough where it really doesn't matter to them since they can afford to lose 50% of their portfolio value and still have enough money, they wouldn't need one.