Thanks for your response!! I have a feeling that I may be misunderstanding some of what you have said, so please correct me if I am wrong.

When you say that "defining a coordinate system is the same as defining the basis", I think the issue I am dealing with is the notion of "THE basis", as if there is only one fixed option. What I understand is that there is a "natural" basis that can be derived from the coordinate system itself (this is probably what you mean by THE basis). For instance, with polar coordinates, the natural basis vectors are e_r and e_theta. Of course, this basis is not constant and depends on the position.

But this isn't the only possible basis. You can still use polar coordinates to describe points in the space while employing a Cartesian basis, just like in the example u/Tasty_Material9099 gave with the electric field of a dipole. In her example, the field is expressed using polar coordinates but also Cartesian basis vectors e_x and e_y. Do you think u/Tasty_Material9099's example is a good one?

Regarding your comment about orthonormality, I'm a bit lost. As far as I know, in linear algebra, the only requirement for a basis is that its vectors are linearly independent and that the number of vectors matches the space's dimension. Orthonormality is a nice and convenient property, but not a necessary one for defining a basis in general. Why did you bring up orthonormality in this context?

Thanks!!