= Solution
Use the nonnegative convention for the <Laplace-Beltrami operator>. For a smooth function $f$ on a <Riemannian manifold> $(M,g)$, three intrinsic definitions are
$$
\boxed{\Delta f=d^*df=-\operatorname{div}(\operatorname{grad}f)=-\operatorname{tr}_g(\nabla df)}.
$$
Here $d$ is the <exterior derivative>, $d^*$ its formal $L^2$ adjoint for the Riemannian volume density, $\operatorname{grad}f$ the <Riemannian gradient>, and $\nabla$ the <Levi-Civita connection>. The bilinear form $\nabla df$ is the <Riemannian Hessian>. The equally common nonpositive convention negates all three definitions simultaneously.
In local coordinates let $g^{ij}$ be the inverse metric <matrix> and $\rho=\sqrt{\det(g_{ij})}$. By the defining relation $g(\operatorname{grad}f,V)=df(V)$,
$$
(\operatorname{grad}f)^i=g^{ij}\partial_jf.
$$
The <divergence of a Riemannian vector field> can be defined by $\mathcal L_V(dV_g)=(\operatorname{div}V)dV_g$. Differentiating the coordinate volume density gives $\operatorname{div}V=\rho^{-1}\partial_i(\rho V^i)$. Thus the second definition becomes
$$
-\operatorname{div}(\operatorname{grad}f)
=-\frac1\rho\partial_i(\rho g^{ij}\partial_jf).
$$
For compactly supported smooth $h$, coordinate <integration by parts> shows
$$
\int_M h\left[-\rho^{-1}\partial_i(\rho g^{ij}\partial_jf)\right]dV_g
=\int_M g^{ij}(\partial_i h)(\partial_jf)\,dV_g
=\langle dh,df\rangle_{L^2}
=\langle h,d^*df\rangle_{L^2}.
$$
Since this holds for every such $h$, the first and second definitions agree. This argument also works on a nonorientable <Riemannian manifold>, using its volume density; no global <orientation> is needed.
For the third definition,
$$
(\nabla df)_{ij}=\partial_i\partial_jf-\Gamma^k_{ij}\partial_kf.
$$
The <Christoffel symbols> satisfy $\Gamma^i_{ik}=\partial_k\log\rho$. This follows by tracing their formula and using $\partial_k\log\det g=\operatorname{tr}(g^{-1}\partial_kg)$. <Metric compatibility> gives
$$
\partial_i g^{ij}+\Gamma^i_{ik}g^{kj}=-\Gamma^j_{ik}g^{ik}.
$$
Substitute both identities into the coordinate divergence expression to obtain
$$
-\rho^{-1}\partial_i(\rho g^{ij}\partial_jf)
=-g^{ij}\partial_i\partial_jf+g^{ij}\Gamma^k_{ij}\partial_kf
=-\operatorname{tr}_g(\nabla df).
$$
This proves all three definitions equivalent. In <geodesic normal coordinates> at a point $p$, the common value is $-\sum_i\partial_i^2f(p)$. Finally, on a closed manifold,
$$
\int_M f\Delta f\,dV_g=\int_M|df|_g^2\,dV_g\geq0,
$$
confirming the chosen <positive Laplace-Beltrami operator> sign convention.
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