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There are many types of backup funds created out of your discretionary income. It's limitless and you can create as much as you want with respect to your demands. There's emergency fund, floating cash, different kinds of savings e.t.c. Emergency fund is just one of them that is dedicated to saving you in the event of uncertainties and unforseen circumstance. You should treat emergency fund as a necessity and not just an option. You can continue accumulating Bitcoin alongside building your emergency fund until it can service one year or more of your expenses as you intend. At this point you can divert the percentage of income set aside for emergency funds into Bitcoin accumulation to increase your aggressiveness having already built out your emergency funds to a feasible extent.
You now focus more on your Bitcoin accumulation alone until you possibly reach your accumulation target or emergency occurs and you need to rebuild the emergency fund.
Your description about how to manage your emergency fund as compared with other kinds of back up funds is confusing, even though maybe you are mostly correct.
It seems worth clarifying that various kinds of back up funds may well have priorities, and truly it could take a year or so just to get to a point of having 3 months of expenses in an emergency fund, depending on if a person might be starting from close to zero and perhaps building a bitcoin investment at the same time.
One of the reasons to put so much emphasis on having an emergency fund when investing into bitcoin is because bitcoin tends to be a very volatile asset, so it cannot serve very well as an emergency fund, even though folks who do not have emergency funds will end up using their investment (in this case BTC) as their emergency fund - which again seems to be a bad practice since it is devolving into a kind of gambling hope that you don't have to tap into your BTC at a time that is not completely of your own choosing, and if any of us might be shooting to invest into bitcoin for 4-10 years or even quite a bit longer, any kinds of tapping into our bitcoin in early stages can really take away from various growth and compounding periods of bitcoin that can come at unexpected times (upward spurts in the BTC price that end up not coming back down to earlier price levels).
I frequently like to assert that we should be striving to have a minimum of 3 months emergency funds that we aspire to never touch, and sure if we are early in our investment into bitcoin, we can build that emergency fund simultaneously towards building our bitcoin, and most likely as we get stronger in our financial cashflow management, then we are going to have additional kinds of back up funds that we would build up and have available to draw into prior to drawing into our emergency fund, so if we are in the earliest of stages of building our emergency fund, we likely cannot be as aggressive in our bitcoin investment since we are fairly vulnerable until building up the emergency fund to at least 3 months, and perhaps we end up needing to have another few months of reserve funds before we might become more aggressive in our bitcoin accumulation - or we might maximize our bitcoin accumulation aggressiveness the more convinced that our back up funds and emergency fund is solid.