$\newcommand{\dd}{\partial}\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Sph}{\mathbf{r}}$A mathematician might denote spherical coordinates by
$$
\left[\begin{array}{@{}c@{}}
x \\
y \\
z \\
\end{array}\right]
= \Sph(\rho, \theta, \phi)
= \left[\begin{array}{@{}c@{}}
\rho\cos\theta\sin\phi \\
\rho\sin\theta\sin\phi \\
\rho\cos\phi \\
\end{array}\right].
$$
The derivative $D\Sph$ is represented by the matrix of partial derivatives
$$
\left[\begin{array}{@{}ccc@{}}
\frac{\dd x}{\dd \rho} & \frac{\dd x}{\dd \theta} & \frac{\dd x}{\dd \phi} \\
\frac{\dd y}{\dd \rho} & \frac{\dd y}{\dd \theta} & \frac{\dd y}{\dd \phi} \\
\frac{\dd z}{\dd \rho} & \frac{\dd z}{\dd \theta} & \frac{\dd z}{\dd \phi} \\
\end{array}\right]
= \left[\begin{array}{@{}c@{}}
\cos\theta\sin\phi & -\rho\sin\theta\sin\phi & \rho\cos\theta\cos\phi \\
\sin\theta\sin\phi & \rho\cos\theta\sin\phi & \rho\sin\theta\cos\phi \\
\cos\phi & 0 & -\rho\sin\phi \\
\end{array}\right].
$$
The (normalized) columns
$$
\frac{\dd\Sph}{\dd \rho} = D\Sph\, \Basis_{\rho},\qquad
\frac{1}{\rho}\,\frac{\dd\Sph}{\dd \theta} = \frac{1}{\rho}\,D\Sph\, \Basis_{\theta},\qquad
\frac{1}{\rho}\,\frac{\dd\Sph}{\dd \phi} = \frac{1}{\rho}\,D\Sph\, \Basis_{\phi}
$$
are the "coordinate vector fields" for spherical coordinates.
In the diagram, the red and blue curves lie in a coordinate surface $\rho = \rho_{0}$ (i.e., a sphere); the blue and green curves lie in a coordinate surface $\theta = \theta_{0}$ (a longitudinal plane); the red and green curves lie in a coordinate surface $\phi = \phi_{0}$ (a cone about the $z$-axis). Small portions of the coordinates curves for $\rho$, $\theta$, and $\phi$ are green, red, and blue respectively. The coordinate fields (not shown) are unit tangent fields along the respective curves.
Generally, if $F$ is an arbitrary change of coordinates (in space, say), then the domain of $F$ is some open subset $U$ of $\Reals^{3}$, and $F$ maps $U$ "diffeomorphically" (smoothly and with smooth inverse) into $\Reals^{3}$. If we write
$$
(x, y, z) = F(u, v, w),
$$
then for each $u_{0}$ we may "restrict" $F$, obtaining a parametric surface $(v, w) \mapsto F(u_{0}, v, w)$. The image of this mapping is the "coordinate surface" $u = u_{0}$.
Similarly, we might consider $(u, w) \mapsto F(u, v_{0}, w)$ for some $v_{0}$, or $(u, v) \mapsto F(u, v, w_{0})$ for some $w_{0}$. The intersection of two coordinate surfaces, say $v = v_{0}$ and $w = w_{0}$, is the "coordinate curve" $u \mapsto F(u, v_{0}, w_{0})$. This parametric curve has a normalized velocity vector at each point, the $u$-coordinate vector field along the curve.
Incidentally, the notation suggests you're reading an engineering text. One cultural difference between engineers and mathematicians is that:
Engineers tend to denote mappings (functions) by assigning letters to input and output values, which can lead to profusions of letters (as in $a, b, c$, $m_{1}, m_{2}, m_{3}$, $r, \theta, \phi$, $R, \gamma, \beta$).
Mathematicians tend to focus on the functional relationship between inputs and outputs, which sometimes goes so far as to actively suppress the names of input and output variables (as in $(x, y, z) = \Sph(\rho, \theta, \phi)$, and speaking of $D\Sph$ instead of the partial derivatives $\dd x/\dd \rho$, etc.)
Among other things, engineering notation tends to be global: Each quantity has a single symbol, sometimes across an entire discipline.
By contrast, mathematical notation tends to be local and context-dependent: The meanings of symbols can change even from paragraph to paragraph, though symbols are chosen to obey loose cultural assumptions. (The beginning of the alphabet ($a$ through $c$) is generally reserved for constants, the middle ($i$ through $n$) for discrete (integer) indices, the late middle ($r$ through $t$ or $w$) for continuous parameters, the end ($x$ through $z$) for coordinate functions.)
Each notational convention has advantages and disadvantages. The better you're able to understand both (even if you spend most of your career in one camp or the other), the easier a time you'll have with the literature (textbooks and papers).
Consider the case of polar coordinates $\langle r, \theta\rangle$ for a point in $E=\mathbb{R}^2$. Then $f()$ here is the function that takes a point in Euclidean coordinates (i.e., $\langle x, y\rangle$) and produces polar coordinates for it; in other words $f(\langle x, y\rangle) = \langle\sqrt{x^2+y^2},\mathrm{atan2}(x,y)\rangle$. Here the $\mathrm{atan2}()$ function is the 'cs version' that's basically $\arctan(x/y)$ but extended in ways that recognize the signs of the coordinates, so that e.g. $\mathrm{atan2}(-x, -y)=\mathrm{atan2}(x,y)\pm\pi$.
Now, this function isn't 1-to-1 on all of $E$; in particular, it's not well-defined at the origin (and there's some fuzziness about coordinates on the x axis). However, $E-\{\langle x,0\rangle: x\geq 0\}$ is a dense, open subset of $E$, and $f()$ is defined on this dense open subset: it takes it to $\{\langle r, \theta\rangle: r\gt 0, 0\lt\theta\lt2\pi\}$, which is an open subset of $\mathbb{R}^2$.
Best Answer
Classically, if $X \subseteq \mathbf{R}^{3}$ is an open set, then a $3$-dimensional coordinate system on $X$ is nothing but an injective, continuously-differentiable mapping $\xi:X \to \mathbf{R}^{3}$ whose differential has rank three at each point. Conceptually, a coordinate system assigns an ordered triple of real numbers to each point of $X$ in such a way that distinct points are given distinct coordinates, satisfying some technical conditions of smoothness. The inverse mapping $\xi^{-1}:\xi(X) \to X$ is sometimes called a parametrization of $X$.
The inclusion map $i:X \hookrightarrow \mathbf{R}^{3}$ is a coordinate system on $X$. If $A$ is an invertible $3 \times 3$ real matrix and $b$ is a vector in $\mathbf{R}^{3}$, then the affine map $T(x) = Ax + b$ defines a coordinate system on $X$. Cylindrical and spherical coordinates are parametrizations of portions of $\mathbf{R}^{3}$; for example, the spherical coordinates mapping (with geographic angles) $$ S(r, \theta, \phi) = (r\cos\theta\cos\phi, r\sin\theta\cos\phi, r\sin\phi),\qquad 0 < r,\quad |\theta| < \pi,\quad |\phi| < \pi/2 $$ parametrizes $X = \mathbf{R}^{3}\setminus \{y = 0, x \leq 0\}$, the complement of a closed half-plane. The spherical coordinates of a point $(x, y, z)$ of $X$ are the numbers $(r, \theta, \phi)$ such that $(x, y, z) = S(r, \theta, \phi)$.
In linear algebra (over an arbitrary field $F$), a "coordinate system" for an $n$-dimensional vector space $V$ takes values in $F^n$, and is usually defined by choosing a basis $B = \{v_{i}\}_{i=1}^{n}$ and mapping a vector $v$ in $V$ to its coordinate vector $[v]_{B}$ in $F^n$. Similarly, in affine geometry one can construct a coordinate system as you describe (by picking an origin and a basis for the resulting vector space).
In the modern point of view, coordinate systems play a secondary role to "overlap maps". For concreteness, let $X$ be an open subset of $\mathbf{R}^{3}$. First, we weaken the criteria for a mapping to be a coordinate system, requiring only that $\xi:X \to \mathbf{R}^{3}$ be continuous and injective. Now assume $\xi_{1}$ and $\xi_{2}$ are "allowable" coordinate systems on $X$ (for some value of "allowable", yet to be determined). An overlap map is a composition $$ \xi_{1} \circ \xi_{2}^{-1}:\xi_{2}(X) \to \xi_{1}(X). $$ The "structure" of $X$ is encoded in the properties of the overlap maps. For example, if the overlap maps are diffeomorphisms, then "smoothness" makes sense for functions $f:X \to \mathbf{R}$: We pick an arbitrary coordinate system $\xi_{1}$ and declare $f$ to be smooth if the composition $f \circ \xi_{1}^{-1}$ is smooth as a function on $\mathbf{R}^{3}$. This definition does not depend on $\xi$, since $$ f \circ \xi_{2}^{-1} = (f \circ \xi_{1}^{-1}) \circ (\xi_{1} \circ \xi_{2}^{-1}), $$ and the overlap map is a diffeomorphism. Similarly, if the overlap maps are affine, then (for example) "line segments" in $X$ make sense: A subset $\ell$ of $X$ is a "segment" if in some (hence every) coordinate system $\xi$, the set $\xi(\ell)$ is a line segment.
Philosophically, a coordinate system is merely how one "transfers" objects and functions on a space $X$ to a objects and functions on a "standard" space, such as $\mathbf{R}^{3}$. The interesting "structure" of $X$ is encoded by the overlap maps, which determine properties of $X$ that are independent of the coordinate system.
When mathematicians speak of a manifold having a smooth structure, they mean that some collection of coordinate systems has been fixed so that the overlap maps are diffeomorphisms. A manifold having an affine structure has coordinate systems whose overlaps are affine (a much more stringent condition). Similarly, one hears of holomorphic, piecewise-linear, and conformal structures, among many others.
To address the question about coordinate vector fields in spherical coordinates: Though the spherical coordinates map parametrizes part of $\mathbf{R}^{3}$ (which is a vector space), the spherical coordinates mapping is not a linear transformation, and therefore does not belong to linear algebra, but instead to multivariable calculus.
The "proper framework" requires some explanation. If $X \subseteq \mathbf{R}^{3}$ is an open set, define the tangent bundle $TX$ to be $X \times \mathbf{R}^{3}$. An element $(\mathbf{x}, \mathbf{v})$ of $TX$ should be viewed as consisting of an element $\mathbf{x}$ of $X$ together with a vector $\mathbf{v}$ "based at" $\mathbf{x}$. It makes sense to take linear combinations of vectors only if they're based at the same point.
In this picture, a vector field on $X$ is a mapping $\Xi:X \to TX$ that assigns, to each point $\mathbf{x}$ of $X$, a vector $\Xi(\mathbf{x})$ based at $\mathbf{x}$. The Cartesian coordinate fields $\mathbf{e}_{i}$ are constant vector-valued functions because Cartesian coordinates change by additive constants under translation (!). By contrast, the spherical coordinate fields are non-constant, because spherical coordinates do not change in a simple way under translation.