Spherical coordinates what are the coordinates of a vector?

Question

I am getting a bit confused about the language for coordinate systems. I have chosen spherical coordinates as an example, but my confusion is general. Let's say we have a vector $$\vec A=A_r e_r+A_\theta e_\theta +A_\phi e_\phi$$ Now from linear algebra I would say that the coordinates of this vector with respect to the given basis is $(A_r, A_\theta, A_\phi)$. Wikipedia among other places, however, says that the coordinates of this system are $(r,\theta,\phi)$. So how do we define the coordinates in a vector space given that these two ideas seem to be different? And wouldn't a better term for $(r,\theta,\phi)$ be the parameters of the coordinate system rather then the coordinates?