Hmmm. You have perhaps unwittingly stated something that gets down to the base point of your misunderstanding. So I will try to use it to get you to understand it once more. You see, in this case, as with many mathematical concepts, the formula is the definition. All those other things you’re looking at are attempts at intuitively explaining the definition (they are not definitions themselves, at least not accurate ones), and they are subject to inaccuracies and imprecision. It’s like explaining something using a metaphor or an analogy – it’s accurate to a certain degree, but then falls apart.
I found this reference:
http://www.stat.cmu.edu/~cshalizi/36-220/lecture-5.pdf
which, in addition to having the mathematical definition of statistical independence, has an interesting and relevant comment:
This may help drive home the point I keep making about the different senses of “independent”.
One final thing I’ll add. It’s interesting how so many people say things like you can’t model the college application process mathematically, that’s why you should apply to a number of reach-match-safety schools. The thing is, that whole idea/strategy is essentially based on a mathematical model. It’s like, reach school are ones where you have less than a 1/3 chance of getting in, match 1/3 - 2/3, safety greater than 2/3 (or whatever probability numbers/ranges you might use), yada yada yada, and using such a strategy increases the chances you’ll get into some number of schools. (Choosing schools based on fit is also an essential factor, and is used on top of this base strategy.)
Some people may not like thinking about things mathematically, it may be hard and difficult to understand. But we still do it all the time.