Why are these two definitions of PLS equivalent?
In general, searching a neighborhood for a better solution and reporting the whole neighborhood are two different things. There are for example some exponential neighborhoods, that is, neighborhoods whose size grows exponentially in the input length $n$, that can be searched in polynomial time for a better solution, see for example A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem by Deineko and Woeginger. The PLS framework, however, is not so much concerned with the running time of searching the neighborhood but with the number of iterations that are needed to find a locally optimal solution.