To achieve the minimum maximum possible weight after moving one item from one box to another, in the second example there is only one option. In the initial distribution of items across the boxes, the weights of the boxes are $5$ and $11$. Consider all the options for moving items between boxes:
\begin{itemize}
\item If you move item number $1$ weighing $4$ from the second box to the first, then the weights of the boxes will be $4+2+3$ and $7$, $\max = 9$.
\item If you move item number $2$ weighing $2$ from the first box to the second, then the weights of the boxes will be $4+7+2$ and $3$, $\max = 13$.
\item If you move item number $3$ weighing $3$ from the first box to the second, then the weights of the boxes will be $4+7+3$ and $2$, $\max = 14$.
\item If you move item number $4$ weighing $7$ from the second box to the first, then the weights of the boxes will be $2+3+7$ and $4$, $\max = 12$.
\end{itemize}
\begin{center}
  \includegraphics[scale=1]{problem_D_new.png} \\
  \small{The first and second example from the problem statement.}
\end{center}