In section 3-4 Do Carmo defines a vector field on a regular surface $S$ as the following:
A vector field $w$ in an open set $U$ of $S$ is a correspondence which assigns to each $p\in U$, a vector $w(p)\in T_p(S)$. The vector field $w$ is smooth (he says differentiable but conflates differentiable and smooth) if for some parameterization $\varphi(u,v)$, and $w(p)=a(u,v)\partial_u\varphi+b(u,v)\partial_v\varphi$, $a$ and $b$ are smooth.
In class we showed that it is insufficient to say a smooth vector field is a smooth map $w:S\to \mathbb{R}^2$ even though $T_p(S)\cong\mathbb{R}^2$ as this implies that there is a smooth parallelizable vector field on $S^2$, and that a proper definition of a vector field requires the notion of the tangent bundle.
However, given that Do Carmo does not mention the tangent bundle, my idea to side step these issues was to define a smooth vector field on a regular surface as follows:
Let $w:S\to \mathbb{R}^3$, with for each $p\in S$, $w(p)\in T_p(S)$, and $w$ smooth as a map $S$ to $\mathbb{R}^3$.
Are there any issue which I am not noticing with this definition?