This is just my current understanding while studying Functional Differential Geometry. This may be completely wrong and may change in the future.
A point on a manifold is an abstract entity representing a location/place on the manifold. When we actually do anything with a point, we use coordinates to represent it. Coordinate functions or charts are used to convert a point into a coordinate tuple. These charts may also have an “inverse”. However the inverse is always used along with another chart in order to get a different coordinate tuple.