In [CEL84, Theorem 1.1, p.489], Crandall, Evans, and Lions give three equivalent definitions of viscosity solution.

As the authors note, the first two are "more appealing in some respects and more convenient for certain purposes" than the third one ([CEL84, Theorem 1.1(iii)]), which was "taken as basic" in the previous paper [CL83].

Q1: In which circumstances is the third definition actually more convenient than the other two?