> And in a general graph where negative edge-weights are permitted, it’s actually impossible to find the shortest path between two nodes without finding the shortest path between the source node and every other node!
If there are negative edge-weights and cycles may occur, there is no shortest path between some nodes. You can keep going through a cycle whose total weight is negative getting "shorter" and "shorter".
It's like a race where one shortcut takes you back in time and leaves you where you started. You can use it to finish the race as far back in time before you started as you want.
Sure, but between "all edges have positive weights" and "there are cycles with negative total weight" there are still graphs that have negative edge weights, yet there's no negative-weight cycle.
If there are negative edge-weights and cycles may occur, there is no shortest path between some nodes. You can keep going through a cycle whose total weight is negative getting "shorter" and "shorter".
It's like a race where one shortcut takes you back in time and leaves you where you started. You can use it to finish the race as far back in time before you started as you want.