I'm trying to understand some basics of calculus on a surface (i.e. a 2-manifold in $\mathbb{R}^3$), without having much knowledge in differential geometry (so from an engineer's perspective). I am not able to get much about how the vector Laplacian on surfaces is defined. There was a similar question here, but the answer there seems to be above my pay grade.
Let's say we have a (sufficiently smooth) surface $S \subset \mathbb{R}^3$, and (sufficiently smooth) functions $u: S \rightarrow \mathbb{R}$, $\vec{v}: S \rightarrow \mathbb{R}^3$. Then, from a lay-man's perspective, for the projection operator(which removes the normal component) $P = \mathbf{I} - \mathbf{n}\mathbf{n}^T$, the surface derivatives can be defined as
\begin{align} \nabla_S u &= P\nabla u = \text{(say) } \left( \begin{array}{c} D_1u \\ D_2 u \\ D_3 u \\ \end{array}\right) \\ \nabla_S \cdot \vec{v} &= P\nabla \cdot \vec{v} = D_1v_1 + D_2v_2 + D_3v_3\\ \Delta_S u &= \nabla_S \cdot \nabla_S u = D_1D_1u + D_2D_2u + D_3 D_3u \end{align}
There seem to be multiple definitions for the Laplacian of a vector field Wiki Page. Two specific ones are what I am trying to understand. The Bochner Laplaican, and the Hodge Laplacian (since those are the ones used in the context I need). There is also a third one here (starting after equation 1.1, till equation 1.3) that has me a bit confused.
Can any of them be understood in a way similar to the surface gradient/surface divergence/Laplace beltrami of a scalar field written above? Is any of them simply a component wise Laplace-Beltrami operator?
\begin{equation} \left( \begin{array}{c} \Delta_S v_1 \\ \Delta_S v_2 \\ \Delta_S v_3 \\ \end{array}\right)? \end{equation}
Another way to look at this is to ask if we can define a Vector Laplacian on a 2-manifold in 3-space without going in to the differential geometry formalism. i.e. using cartesian coordinates only.