I'm getting a bit confused about a conceptual question. Consider a differentiable manifold (e.g. a 2-sphere)...reading the Wiki article, it seems that we'll end up with defining different types of charts whenever we look at the manifold from the extrinsic or the intrinsic viewpoint. However, reading other sources, there's no such difference.
Moreover, even if the intrinsic approach is usually described as "coordinates-free", it however relies on the definition of coordinate charts (and transition maps).
Where am I lost?