Understanding the heuristic used for approximate string searching through an FSA
Those two statements are correlated: "giving priority to the highest word position" is the same as "the node that has advanced farthest in the word is given priority". 

This appears to be why the algorithm selects state three. State zero does not result in an advancement in word position, whereas the transition to state three is further in the word position and so satisfies their heuristic of highest word position and thus gets priority for first expansion. 

To see why this heuristic results in better results, examine the node zero. Deleting 'd' you see all the letters remaining are in the reachable nodes from zero, which means $h = 0$, and we only performed 1 operation, so $f = 1$. However in order to get to the acceptance state along the shortest path we have to take another operation to go to 3 while adding 'c' back, then we transition through the states 5, and 7 making no changes. Obviously the fastest (and shortest) path is to simply swap 'd' to 'c' and transition to state three first, which the heuristic suggests in this case.