Analysis of updating vertex and one example?
In Dijkstra's algorithm there are $|E|$ update operations and $|V|$ remove_min operations.
To implement the algorithm, we can employ different types of data-structures. Since you are looking for the best update cost per vertex, you can employ the Fibonacci heap or Hash map implementation since they give you the minimum update cost (as per the table mentioned in your question). Now, suppose you employ Fibonacci heap. Then, the total amortized update cost for Dijkstra's algorithm would be $|E| \cdot O(1) = O(|E|)$. Therefore, the amortized update cost per vertex is: $O(|E|/|V|)$.

These are the three main implementations of Dijkstra's algorithm:

Array Implementation (a.k.a. Hash Map as per your table): Update $O(1)$ and Remove_Min $O(|V|)$

Binary Heap Implementation (a.k.a. Hybrid Binary Heap as per your table): Update $O(\log |V|)$ and Remove_Min $O(\log |V| )$

Fibonacci Heap Implementation: Update Cost $O(1)$ and Remove_Min Cost $O(\log |V|)$


Note that Fibonacci Heap provides the best update cost and the best remove_min cost among these three options. Therefore, it is the best of both worlds. Hence, it is best for Dijkstra's algorithm and you can remember this implementation only.