Seeking references regarding Zero Extension of Sobolev ($W_0^{1,2}$) functions on a compact domain on $C^\infty$ Riemannian manifold

Since we define $W^{1,2}_0(\Omega)$ to be the closure of $C^{\infty}_0(\Omega)$ (smooth compactly supported functions in $\Omega$), the assertion easily follows since $C^{\infty}_0(\Omega)$ can be extended by zero.

More precisely, given $f \in W^{1,2}_0(\Omega),$ let $f_n \in C^{\infty}_c(\Omega)$ be a sequence such that $f_n \to f$ in $W^{1,2}(\Omega).$ Then extending each $f_n$ by zero to $M$ gives $\tilde f_n \in C^{\infty}_0(M) \subset W^{1,2}(M),$ since we have $$ \lVert \tilde f_n - \tilde f_m \rVert_{W^{1,2}(M)} = \lVert f_n - f_m \rVert_{W^{1,2}(\Omega)} \to 0$$ as $n, m \to \infty,$ we have $(\tilde f_n)$ is Cauchy in $W^{1,2}(M).$ Hence by completeness of $W^{1,2}(M)$ there is a unique limit $\tilde f \in W^{1,2}(M).$ Further (by $L^2$ convergence) passing to a subsequence such that $\tilde f_{n_k} \to \tilde f$ pointwise almost everywhere, we see that $\tilde f = f$ almost everywhere in $\Omega$ and $\tilde f = 0$ almost everywhere in $M \setminus \Omega,$ so $\tilde f$ is the zero extension of $f$ to $M.$

Note this does not make use of the trace operator, and in fact holds for any $\Omega \subset M$ open.

For a bounded Lipschitz domain $\Omega \subset \Bbb R^n,$ it is well-known (see e.g. Evans Ch. 6, or any text on Sobolev spaces) that $W^{1,2}_0(\Omega)$ coincides with the space of $f \in W^{1,2}(\Omega)$ with zero trace. If we use this to define $W^{1,2}_0(\Omega)$ then one needs to do a bit more work; we can work locally and reduce to the flat case as mentioned in the comments. Note the the precise argument in this case will depend on how exactly you define $W^{1,2}(M),$ and is closely related to the construction of the trace operator (which is typically done by flattening the boundary and approximation by smooth functions).