I'm reading a physics text which dives into some mathematical concepts that are new to me. Consider a function $F(q,\dot q)$ (dynamical variable) where here $q$ and $\dot q$ represent sets of local generalized position coordinates $q^\alpha$ and velocity coordinates $\dot q^\alpha$. A dot above indicates a total time derivative ($\dot F = dF/dt$). Thus,
$$\dot F (q , \dot q) = \frac{\partial F}{\partial q^\alpha} \dot q^\alpha + \frac{\partial F}{\partial \dot q^\alpha} \ddot q^\alpha.$$
Here, the Einstein summation convention is assumed, implying a summation over the index $\alpha$. In general, the equations of motion for a particular physical system provide a function $\ddot q^\alpha = \ddot q^\alpha (q, \dot q)$, so the above equality is indeed strictly a function on only $q$ and $\dot q$.
Now, my textbook generalizes this derivative into an operator $\Delta$, which send functions on $(q, \dot q)$ to other functions on $(q, \dot q)$.
$$\Delta = \dot q^\alpha \frac{\partial}{\partial q^\alpha} + \ddot q^\alpha\frac{\partial}{\partial \dot q^\alpha}$$
This allows us to write
$$\dot F = \Delta F.$$
My textbook proceeds to refer to $\Delta$ as a vector field and additionally generalizes the notion of a vector field $X$ on $(q, \dot q)$ as
$$X = X^\alpha_1 \frac{\partial}{\partial q^\alpha} + X^\alpha_2 \frac{\partial}{\partial \dot q^\alpha},$$
where the $X_1^\alpha$ and $X_2^\alpha$ are functions on $(q, \dot q)$.
My current understanding of the term vector field is something like an electric field, where every point in a space is associated with a vector. Can anyone explain to me how these two notions of vector field are related? Why is my textbook referring to $X$ and $\Delta$ as vector fields when they act as operators and are not vector-valued?