Add a road to a network (Graph problem with Dijkstra)
Who knows. You'd have to ask them; that doesn't provide enough details to narrow it down.

If I had to guess, I imagine that perhaps they had in mind that they would do Dijkstra's algorithm once to measure the distance from $s$ to each vertex; then for each candidate edge $(u,v) \in E'$, they would use Dijkstra's algorithm to find the distance from $u$ to $t$ assuming $(u,v)$ has been added to the graph; and use this to find the best edge to add. 

In any case, their alternative solution is considerably slower, so it's a poor choice and should not be used.  It's far faster to use the first solution you listed.