We had a bottle of wine between the three of us. I knew that. But still, where's the mistake there?
The mistake there is obviously in not having one each!
The case of 'between' and 'among' is an excellent example of textbook writers come across an inadquate explanation, and then try and make the language fit the explanation.
The difference between 'between' and 'among' is that 'between' is used for discrete entitities and 'among' for an amorphous mass.
Now the important thing is how you are visualizing what you are describing. If you say
He walked between the crowd the image is of his path cutting the crowd into two or more discrete parts, rather like Moses crossing the Red Sea. if you say
He walked among the crowd then we get the idea of a pleasant amble in the midlst of a lot of people.
We say
They haven't got a decent idea between them because we are considering each their cluelessness individual by individual, but
There is a general consensus among them because we are considering them as a whole.
The idea that "between" can be only used for two tnings is a particulary pernicious piece of misinformation.