Two definitions of Safe Edge
They are two different definitions.
The interview definition calls a safe edge one that is not part of any cycle and therefore cannot be removed from $G$ without disconnecting it, thus changing the resulting MST.
Notice that this definition depends solely on the chosen edge and on $G$. Notice also that it does not depend on the edge weights in any way.
The definition in the notes defines a safe edge $e$ as one that can safely be added to a subgraph $A$ of $G$ so that $A + e$ is a partial MST of $G$.
In this case $A$ is probably thought to already be a partial MST of $G$, since otherwise no edge would be safe, although this is not formally required.
Notice that this definition depends on the particular choice of $A$ and on the edge weights of $G$.
As an example, let $G$ be a cycle on $n$ nodes, where the $n$ edges have weights $1, \dots, n$. None of these edges is safe according to the interview definition.
If $A$ is the empty graph then all edges except for the one of weight $n$ are safe w.r.t. the definition in the notes.
If $A$ is the subgraph containing the edge of weight $n$ then no edge is safe w.r.t. the definition in the notes.