I'm reading "A Visual Introduction to Differential Forms and Calculus on Manifolds", specifically the section on equivalence of directional derivatives to vectors acting as operators on functions defined on the manifold, which in turn are equivalent to differentials of functions acting on vectors.
I can't wrap my head around the notation, probably since it's my first time learning the subject. So first we define $df(v_p)$ as the directional derivative of $f:\mathbb{R}^3\to\mathbb{R}$ evaluated at $p$. This leads us to conclude (for example on $\mathbb{R}^3$): $$df(v_p)=\sum_{i=1}^3\frac{\partial f}{\partial x_i}v_i$$
Once we replace $f$ by cartesian coordinate functions (let's call them $x,y,z$), then we get $dx(v_p)=v_1$ (and similarly for the other 2 coordinate functions). And then for the basis $\{e_x,e_y,e_z\}\in T_p(\mathbb{R}^3)$ we can conclude that $dx(e_x)=1, dx(e_y)=dx(e_z)=0$ and similarly for $dy$ and $dz$, which allows us to identify these with the cotangent basis vectors.
Why don't $dx,dy,dz$ refer to any specific point in the manifold? Do I interpret them to mean that they're sections of the cotangent bundle that pick out the cotangent space basis for every point $p$ (i.e. $\{dx_p,dy_p,dz_p\}$) and it is these specific basis elements that act on the corresponding tangent space basis vectors as per the Kronecker delta rule?
Why do we choose the $df$ notation? For example in the equation above, the LHS is the infinitesimal increment in $f$ (starting from point $p$) as we move in the $v_p$ direction. The RHS, on the other hand, is the slope of the tangent surface at $f$ at point $p$ where the slope is evaluated in the $v_p$ direction. Aren't the slope and infinitesimal increment in $f$ supposed to be different things?
Why couldn't we just replace $df$ by some operator $\lambda(f)$ that acts on a vector $v_p$ to give the slope of $f$ in $v_p$ direction? Why do we instead choose to go with a (seemingly) inconsistent notation?