The Fig. 5 in the question depicts AC-algorithm with backtracking.
In fact, without backtracking, checking arc-consistency alone can hardly solve the 4-Queen problem. Unless it happens that choices made so far will be extended to a desired configuration, any algorithm to solve CSP have to backtrack so as to check other possibilities.
Here is a relevant excerpt from the article.
Conversely, if it does not become possible to place a Queen at the pointed column, like in Fig. 5(c), backtracker will backtrack the previous assigned Queen’s column to reassign the conflicted Queen to a newly safe cell at the same column. Then, it will, at first, clear the constraints it created by calling the implemented modular function.