The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in
Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819
Here are a few details. Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.
For any $u\in C^\infty(E)$ we have
$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$
Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a connection $\nabla$ on $E$ compatible with the metric on $E$ such that
$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$
where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.
If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce
$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$
At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.