What is the difference between a parameterization of a manifold and a local chart?

Question

I am confused on the difference between a parameterization of a manifold $M$ and local charts on $M$. If $M$ has dimension $n$, we may find a subset $U \subset \mathbb{R}^n$ such that there exists a homeomorphism $X: U \rightarrow M$. The map $X$ is said to be a parametrization of $M$. Thus it appears to me that the points in $U$ give us coordinates on $M$. If so, how is this any different than a coordinate chart $V$ on $M$? Secondly, wouldn't the existence of such a homeomorphism mean that $M$ is trivial (can be covered by a single chart)?