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?
(i). The semijet definition [CEL Theorem 1.1 (i)] is the natural way to derive comparison principles for PDEs using, e.g., the Crandall-Ishii lemma [CIL, Theorem 3.2].
(ii). Here are some various examples in which [CEL Theorem 1.1 (ii)] is easier to work with than the above:
(iii). [CEL Theorem 1.1 (iii)] is motivated by maximums of products of a solution with a test function. Precisely, let $u$ be a classical solution and $\phi$ be continuously differentiable. Then if $\phi(y)u(y) = \max\{\phi u\}>0$, the product rule yields $$(D(\phi u))(y)=\phi(y)Du(y)+u(y)D\phi(y)=0.$$ Moving some terms around, $$Du(y) = -\frac{u(y)}{\phi(y)} D\phi(y).$$ The version of (iii) appearing in [CEL] is the same idea, except with $\phi(y)(u(y)-k)$ being a positive maximum.
Perhaps a disappointing answer: I cannot think of a situation in which (iii) would be more convenient to work with than (ii). As stated in [CEL], "L. C. Evans has observed that the criterion utilizing extremals of $u - \phi$ is more convenient in various situations." Moreover, I have not seen (iii) in any recent works. It is possible that the (great) authors exercised extra caution in writing "various situations," when perhaps they might have meant "all situations we can think of." However, I cannot speak for them :-)
