If you think about it, it's really all one big pot of money. The idea behind an "emergency fund" is that you want to make sure your financial life has stability: it's not going to be suddenly driven into the red, below $0. As long as that doesn't happen, you can figure out how to live your life as you want.
The reason we separate out an "emergency fund" is to simplify decision making. In theory, every single purchase you make should include a consideration of how it destabilizes you. Every $100 you spend on groceries is $100 you won't be able to bring to bear if you get fired or have a major accident. In practice, this would be a crippling way of thinking about things. You don't know what emergencies can hit you, nor when they will hit. That's why they're "emergencies." If you had to think about them all the time, it'd be horrible! You would end up simply not thinking about it (like most people), and then the emergency hits when you don't have enough cash to stay solvent.
The purpose of an "emergency fund" is to help make these decisions easier. If you have money set aside for "emergencies" that you only have to think about every now and then, you can make the decisions in the rest of your financial life without too much concern for them. You don't have to worry about that $100 in groceries because you are confident that if an emergency hits, that $100 won't be the straw that broke the camel's back because you have reserves to draw on.
So you should define an "emergency fund" in a way which is most helpful for you to remain stable and solvent without having to fret about it too much. For most people, the criteria for tapping that fund is very high, because the goal is to not have to think about it all that much. If you wanted to, you could feel free to lump those "medium predictability" items into the emergency fund, but it just means you have to spend more time and effort thinking about the state of the fund. Every medium predictability purchase has to come with the thought process "what is the state of the emergency fund? Could this purchase meaningfully destabilize my ability to handle emergencies?" Your emergency fund might yo-yo under these extra purchases, which could force you to think about the state of your emergency fund for normal purchases. That'd be bad.
Different people might want to think about things different ways. I'm a big-picture guy, so I prefer to think about all of my assets as one big account when I make a lot of my decisions. My wife, on the other hand, prefers not to have to think that way when she makes her purchases. For her, having a very discrete "emergency fund" has great value. For me, it has less. So when I look at the finances, I choose to lump the emergency funds in with, say, the funds to re-do our backyard (something we are looking at doing over the next 2-5 years). For me, that is the most natural way to deal with analyzing the risks -- I just have to be aware of how backyard purchases interact with our safety net. My wife prefers to keep those funds separate in her head, so that she can look at how to spend money on the backyard without thinking about how it affects our emergency readiness. While complicated, it shows that even within a household, it's possible to think about emergency funding two different ways. (it causes minimal headaches, though a fair bit of book-keeping)
So define "emergency fund" however suits you and your life best. However, practically speaking, most people find it desirable to not put those medium predictability purchases into the same bucket as emergencies. Those that do find it desirable to put them in the same bucket typically have a personal reason for why that suits their needs better.