The following is the definition that I'm confused about:
Suppose $S\subseteq M$ is an immersed $k$-dimensonal submanifold. A local parametrization of S is a continuous map $X: U\to M$ whose domain is an open subset $U\subseteq \mathbb{R}^k$, whose image is an open subset of $S$, and which, considered as a map into $S$, is a homeomorphism onto its image.
By "considered as a map into $S$," does Dr. Lee mean that $X(U)$ is homeomorphic to a subset of $S$, where $S$ has its submanifold topology or the subspace topology with respect to $M$ (since $X$ has codomain $M$)?
It means that if you consider the map $X: U \rightarrow X(U)$, this is a homeomorphism if you give to the set $X(U)$ the subspace topology inherited by $S$. While, $S$ has the topological structure given by the immersion, since it is an immersed submanifold.