Briefly, I'm asking why commuting the metric and the second fundamental form should make a difference.

Normally I wouldn't think much of this, but I came across the issue in the paper "Hypersurfaces of constant curvature in hyperbolic space. II." by Guan and Spruck. They in turn cite another paper, "Nonlinear second order elliptic equations IV. Starshaped compact Weingarten hypersurfaces." by Caffarelli, Nirenberg, and Spruck, which, sadly, is not easy to access. I found it strange that such a seemingly basic fact needed such a "high powered" reference to be addressed.

For example, consider a function $f:\mathbb{R}^2\to\mathbb{R}$ and let $S = (x,y,f(x,y))$ be the graph of $f$ in $\mathbb{R}^{3}$. Then the induced metric on the surface is $g_{ij} = \delta_{ij}+f_{i}f_{j}$ and the second fundamental form is $\frac{f_{ij}}{\sqrt{1+f_{1}^{2}+f_{2}^{2}}}$. The shape operator $s$ should then be $s_{j}^{i} = g^{ik}h_{kj}$. However, when I take the trace of $s$ (as a matrix) I obtain $\frac{f_{11}(1+f_{2}^{2})+f_{22}(1+f_{1}^{2})}{(1+f_{1}^{2}+f_{2}^2)^{3/2}}$. Note that this is not the expression that arises in the minimal surface equation ... but it should be, because it's the trace of the second fundamental form with respect to $g$! (At least, that's what I would believe by reading geometry texts, such as Lee, "Riemannian manifolds: an introduction to curvature").

However, let $\gamma_{ij} = \delta_{ij}+\frac{f_{i}f_{j}}{1+\sqrt{1+f_{1}^{2}+f_{2}^{2}}}$, so that $\gamma_{ij}$ is the square root of $g_{ij}$ via $\gamma_{ik}\gamma_{kj} = g_{ij}$. Consider the operator $t = \gamma^{ik}h_{kl}\gamma^{lj}$ ... in other words, $t$ is $s$ but after commuting a square root of $g_{ij}$. Then if I take the trace of $t$, as a matrix, I obtain $\frac{(1+f_{1}^{2}+f_{2}^{2})\Delta f - f_{ij}f_{i}f_{j}}{(1+f_{1}^{2}+f_{2}^{2})^{3/2}}$, which is exactly the expression arising in the minimal surface equation.

What is going on here? The idea of taking the trace after commuting a square root power seems strange. None of the geometry books I've looked at address this, and indeed in my time studying geometry I've never seen this issue arise before. Any help (even just a suggestion for a reference) would be appreciated.