Consider a $1$-dimensional smooth compact manifold with boundary $M$. Since $M$ is $1$-dimensional, then $\partial M$ is a $0$-dimensional manifold and it will have an induced orientation. I want to clarify some things about this. An orientation for a $0$-dimensional manifold is simply a function $\epsilon:\partial M\rightarrow \{\pm 1\}$ . Now given an outward-pointing vector field $Y$ of $M$ is the induced orientation $\epsilon$ on $\partial M$ the one such that $\epsilon \cdot Y$ is positively oriented with respect to the orientation of $M$. From the definitions of induced orientation that I found, from the book "Introduction to manifolds" I belive this will be it but I wanted discuss this with more detail.
Any insight is appreciated, thanks in advance.
If $Y$ is a (non-vanishing) outward-pointing vector field on $\partial M$, it defines an orientation on $\partial M$ exactly as you outlined: for $p \in \partial M$, define $\varepsilon (p) = 1$ if and only if $(Y(p))$ is a positively oriented basis of $T_pM$ with respect to the orientation of $M$.