Local parametrizations and coordinate charts on manifolds

Solution 1:

1. A manifold is an abstract axiomatic notion. Therefore, points in the manifold $M$ may be very weird objects. They may be matrices, other manifolds, whatever you want. As you say, the charts allow us to label points in the manifold by actual points in actual euclidean space, with the goal of doing some geometry and calculus (which we don't know how to do in $M$ a priori). In this sense, the coordinates of a point $p\in M$ are the corresponding coordinates of the point in $\mathbb{R}^{n}$, and the parametrization of the points in $M$ is this process of labelling the points in $M$ by points in space, so it is the inverse of the chart map.

2. You are right, if your chart assings to a point in an open subset of the sphere the two angles say with a meridian and the equator, then the coordinates of the point are those angles (a point in $\mathbb{R}^{2}$, as you said), and the parametrization is the inverse image of this map, which then assigns to points in the plane points in the sphere. Sorry for being circular, but indeed, it parametrizes points in the manifold by points in the plane (in the broad sense of the word parametrizing, e.g. points in projective space parametrize 1-dimensional linear subspaces of the underlying vector space).

3. I don't agree with your conclusion here. As I mentioned above, the goal of parametrising the points in your abstract manifold by points in euclidean space is to be able to do some geometry and calculus. Therefore, labelling points like that can always be done, at least in order to get a local parametrization of points in your manifold, which can already be very useful. Another example: you can do differential calculus on smooth manifolds thanks to these parametrizations, even if $M$ is not locally flat. If your point is something like "the manifold looks locally like euclidean space if it is locally flat", then I also disagree: this is to me a subjective statement, which depends on the properties that you are considering. For example, a locally flat manifold $M$ may very well not be a vector space. But euclidean space is. So the question is always relative to the properties you are looking for at that moment.