Your two examples are of the following form: Fix a collection $C$ of subsets of $n$ such that any two elements of $C$ have non-empty intersection and any other subset of $n$ either contains or is disjoint from an element of $C$. Then let $A$ be the collection of those subsets of $n$ which contain an element of $C$.
In the first example $C=\{\{0\}\}$ while in the second example $C$ is the collection of subsets of size $(n+1)/2$. It is not hard to show that any example must be of this form.