# Local search (Hill Climbing) scope and definition

I'm taking an artificial intelligence class and in one of the recent lectures the topic was local search algorithms, more specifically Hill Climbing. At one point the professor showed the classic 8-queens puzzle to illustrate the algorithm working, he stated that in it a queen can only move one row up or down, so for example in a column with the following configuration:

In the first iteration the queen can only move to either rows 4 or 6 (8 rows, 0-based index), then in the next iteration it would be stuck in a local maxima because neither row 3 nor 7 have values lower than 14, so the algorithm would then proceed to the next column or terminate. He argued that even tough a queen can legally move directly through all the rows of its column doing so would make the algorithm fall of the concept of "Local search", because a local search move is limited to only one of its closest neighboors per time/iteration.

I found this weird because it appears to me that such a limitation greatly reduces the usefulness of hill climbing (in the 8-queen problem it makes the success rate abysmal). So I would like to know what's the correct/formal definition of local search and hill climbing, should it always be constrained to the closest neighboor? what constitute a closest neighboor? I would especially appreciate some book references.