I believe as a general rule...
A real emergency fund is 3-6 months of expenses, the "mini 1k e-fund" is just a starter to help while debt is paid off.
I believe in questioning "general rules", especially when those general rule are put in place by those that profit from them (banks able to earn money on your 3-6 months expenses sitting in a zero interest checking account for example). The discussion forums for the course are overwhelmingly saying keep it in cash (not even in a bank, but cash under the mattress).
For the record, my personal opinion is that an emergency fund should be multi-tiered: 1k in cash/checking for things you can't cash flow/couldn't plan for (this also happens to be the deductible on all of my insurance plans, which gives me a little extra piece of mind), 3-6 month expenses invested in a manner befitting your particular risk factor (Roth IRA in index funds being my preference, but YMMV).
I'm also of the school of thought that very few things constitute as an emergency if you put some thought into it.
Lose my job? Pick up more hours at my side job, which supplemented with my rental income should sustain me on a bare bones spending level. I have job loss savings, but because I have back up plans in place I feel comfortable having those funds invested.
Major car repair? Grow some balls and sell the car. Replace with a scooter for the trips I am not comfortable doing by bike.
Major home repair? I have that 1k insurance deductible set aside for catastrophic events. Monitoring the state of the home should give me enough warning to save for any time-damage repairs. If not? Just save a little less until I pay it off.
Injury or sickness? Insurance then dip into the invested savings if necessary. This is the only one that can really blindside a person, but it's also the one least worth worrying about in my opinion. It will either happen or it won't: if I have to pay a penalty on tax-advantaged funds to have a live-saving surgery so be it.
In all of these, the "mini 1k efund" plays a role. I think it is more about getting out of debt. 1k is sufficient to handle most "emergencies" when combined with other directed savings, hence why I keep mine in a savings account.