Hi,
As you know, I was trying to offer a general principle that might be useful. However, there are lots of special contexts and idiomatic situations . This is what makes it very hard for native speakers to explain articles clearly. However, let me try a few comments on your latest example.
You could use 'a set' rather than 'the set', without really changing the meaning.
The 'the' stresses that there is only one set of all tea-cups, and this uniqueness of each set is the point of the whole paragraph.
There is only one set of all tea-cups. But, there can be 500 lists of the four directions. Or maybe no list.
Articles are tricky, aren't they?
Best wishes, Clive