This is a nice question. In my view, the choice of test function is motivated by the idea of taking some positive quantity that solves an elliptic equation, and looking at the equation for its logarithm. The new equation often contains terms with a helpful sign. This idea is pervasive in geometry and elliptic PDEs, and some examples are below.
(1) The basic case to consider is that $u$ is positive and superharmonic. Then $v := -\log u$ satisfies $|\nabla v|^2 \leq \Delta v$, which gives local universal bounds on the integral of $|\nabla v|^2$ after multiplying by standard cutoffs and integrating by parts.
(2) In your context, the choice of test function $H(u)$ satisfies $\nabla u \cdot \nabla(H(u)) = |\nabla (V(u))|^2$, where $V(u) = c_1\log(c_2 - u)$ with $c_2 - u$ positive. I view the estimate as coming from integrating the equation for $V(u)$. To illustrate how this works in a simple context, assume that $u \in C^2_0(B_1)$ satisfies $\Delta u \geq -1$, and that $u \leq M$. Then for $w := M+1-u$ we have that $v := -\log(w)$ satisfies $|\nabla v|^2 \leq 1 + \Delta v$. Thus the integral of $|\nabla v|^2$ is bounded by $1$.
(3) The Bombieri-De Giorgi-Miranda interior gradient estimate for a solution $u$ to the minimal surface equation is based on the fact that the vertical component $\nu^{n+1}$ of the unit normal to the graph of $u$ is positive and superharmonic (on the graph). The proof uses the equation for $v := -\log(\nu^{n+1})$, which just as above contains a useful term quadratic in $|\nabla v|$.
(4) The Li-Yau proof of the Harnack inequality for a harmonic function $u$ is obtained by looking at the quantity $w := |\nabla (-\log u)|^2$. The key is that $w$ solves a differential inequality with the powerful term $\frac{2}{n}w^2$, which allows one to bound $w$ from above locally by a universal constant.
I am sure there are other interesting examples, and I am not sure where the first instances of the "log" trick appeared.