Clarification sought for definition of a cut that respects a set A of edges in Graph Theory
One mistake you're making with the statement:


  In every example in the text, the set A is coincident with one of the partitions of the cut but I cannot see why this must be.


$A$ is a set of edges, while a cut is a partition of vertices: $(S, V - S)$.

Suppose, as you say, that $A$ is a subset of the set $\{(u,v): u,v \in S\}$, then $A$ respects the cut $(S,V-S)$ because none of the edges with both ends in $S$ will cross the cut.

If you add a new vertex $v'$ into $S$, and some other edges from $v'$ to other vertices $x$ in $S$, then $A$ can include ALL of these edges and still respect the cut $(S,V-S)$ because both $v'$ and the other edges from $v'$ are in $S$.