I think going through a city does not mean entering the city.
A counter case for Case 2 would be :
n = 4, m = 3
val_to_enter = {1000, 500, 1, 4}
edges :
1 2 2
2 3 1
3 4 1
now for city 1 according to you the answer would have had been = 502(entering city 2) .
Instead a optimal way would have had been to enter city 3 with cost = 6.