Consider a system of $N$ material points described by position vectors ${\bf x}_1, \ldots, {\bf x}_N$ in a reference frame ${\cal R}$. These position vectors are not free to assume any configuration in the physical space, but they are constrained to satisfy some constraints which, possibly, may depend on time,
$$f_j({\bf x}_1, \ldots, {\bf x}_N, t) =0\quad j=1,\ldots, c < 3N\:.\tag{1}$$
If these functions are smooth and satisfy a condition of functional independence (I do not want to enter into the details), we can choose $n:= 3N-c$ abstract coordinates $q^1,\ldots, q^n$ which can be used the embody the constraints into the formalism. This result holds locally around every admitted configuration and around a given time $t$.
As a matter of fact, we can locally (in space and time) represent the position vectors ${\bf x}_1, \ldots, {\bf x}_N\quad i=1,2,\ldots, N$ as known functions of the said free coordinates.
$${\bf x}_i = {\bf x}_i(q^1,\ldots, q^n,t)\tag{2}$$
When $q^1,\ldots, q^n, t$ varies in their domain (an open set in $\mathbb{R}^{n+1}$), the vectors ${\bf x}_i(q^1,\ldots, q^n,t)$ automatically satisfy the constraints (1). The admissible configuration are therefore determined by the free coordinates $q^1,\ldots, q^n$ at each time $t$.
Now we pass to the notion of virtual displacement compatible with the set of constraints (1). It is defined by fixing $t$ and computing the differential of the functions ${\bf x}_i$ as functions of the remaining variables. The virtual displacement of the system at time $t_0$ around a permitted configuration determined by $q_0^1,\ldots, q_0^n$ is the set of $N$ vectors in the real space
$$\delta {\bf x}_i = \sum_{k=1}^n \left.\frac{\partial {\bf x}_i}{\partial q^k}\right|_{(q^1_0,\ldots, q_0^n,t_0)}\delta q^k\:, \quad i=1,\ldots, N\:.$$
Above the numbers $\delta q^k\in {\mathbb R}$ are arbitrary, not necessarily "infinitesimal" (which does mean anything!).
Example
Consider a point of position vector ${\bf x}$, constrained to live on a circle of radius $r=\sqrt{1+ct^2}$, where $c>0$ is a known constant.
This circle is centered on the origin and stays in the plane $z=0$.
Here we have just two constraint
$$f_1({\bf x}, t)=0, \quad f_2({\bf x}, t)=0$$
where
$$f_1({\bf x}, t) := x^2+ y^2+ z^2 - (1+ct^2)\:, \quad
f_1({\bf x}, t) := z $$
if ${\bf x}= x{\bf e}_x+ y{\bf e}_y+ z{\bf e}_z$
Locally we can use, for instance, the coordinate $q^1=x$ to describe a portion of circle and we have
$$x = q^1\:, \quad y = \sqrt{(1+ct^2) - x^2}\:, \quad z=0\:.$$
The relations (2) here read
$${\bf x}(q^1,t)= q^1 {\bf e}_1+ \sqrt{(1+ct^2) - (q^1)^2}{\bf e}_2$$
The virtual displacements at time $t_0$ are the vectors of the form
$$\delta {\bf x} = \delta q^1 {\bf e}_1+ \frac{\delta q^1}{\sqrt{(1+ct_0^2) - (q^1)^2}}{\bf e}_2$$
for every choice of $\delta q^1$.
The geometric meaning of $\delta {\bf x}$ should be evident: it is nothing but a vector (of arbitrary length) tangent to the circle at time $t_0$ emitted by a configuration determined by the value $q^1$.
REMARK It is worth stressing that varying $t$, the circle changes! Virtual displacement are defined at given time $t$.
The discussed example is actually quite general. The virtual displacement are always vectors which are tangent to the manifold of admitted configurations at given time $t_0$.
The prosecution of Lagrangian approach (once assumed the validity of the postulate of ideal constraints and introducing interactions for instance defined by a Lagrangian) consists of finding the evolution of the system not in terms of curves $${\bf x}_i = {\bf x}_i(t)\:, \quad i=1\,\ldots, N$$ in the physical space. The motion is described directly in terms of free coordinates, i.e., curves $$q^k=q^k(t)\:, \quad k=1,\ldots, n$$
Just at this level it makes sense to introduce the notation $\dot{q}^k = \frac{dq^k}{dt}$, because here we have a curve $t$-parametrized describing the evolution of the system. Before this step $q^1,\ldots, q^n$ and $t$ are independent variables.
A posterori, if we have a description of the motion of the system in terms of free coordinates, we also have the representation in the physical space by composing these curves with the universal (= independent of any possible motion) relations (2),
$${\bf x}_i(t) = {\bf x}_i(q^1(t), \ldots, q^n(t), t)\:, \quad i=1\,\ldots, N\tag{3}$$
It is finally interesting to compare virtual displacements with real displacements when we have a motion $$q^k=q^k(t)\:, \quad k=1,\ldots, n\:.$$
In the physical space, the velocities with respect to the reference frame ${\cal R}$ are given by taking the derivative with respect to $t$ of (3),
$${\bf v}_i(t) = \sum_{k=1}^n \frac{\partial {\bf x}_i}{\partial q^k}\frac{dq^k}{dt} + \frac{\partial {\bf x}_i}{\partial t}\:.$$
An approximate displacement ascribed to an interval of time $\Delta t$ is
$$\Delta {\bf x}_i = {\bf v}_i(t)\Delta t = \sum_{k=1}^n \frac{\partial {\bf x}_i}{\partial q^k}\frac{dq^k}{dt}\Delta t + \frac{\partial {\bf x}_i}{\partial t}\Delta t\:,$$
which can be rephrased to
$$\Delta {\bf x}_i = \sum_{k=1}^n \frac{\partial {\bf x}_i}{\partial q^k}\Delta q^k + \frac{\partial {\bf x}_i}{\partial t}\Delta t\:. \tag{4}$$
This identity has to be compared with the definition of virtual desplacement
$$\delta {\bf x}_i = \sum_{k=1}^n \frac{\partial {\bf x}_i}{\partial q^k}\delta q^k\:.$$
Even if we choose $\delta q^k = \Delta q^k$, the right-hand sides are different in view of the term $ \frac{\partial {\bf x}_i}{\partial t}\Delta t$ which accounts for a part of the displacement, in real motion, due to the fact that constraints may depend on $t$ explicitly, as in the example above.
Best Answer
I see your question can be expressed in words as "when the virtual displacements/velocities agree with the allowed ones?" that's, as you said,
$ \frac{\partial \mathbf{x}_{i}}{\partial t} = 0 $
that is to say that the position vector $r$ is expressed in terms of $q_{k}$'s only and doesn't contain $t$ explicitly, so as the constraints. i.e. the system is scleronomic .
example of this system is the pendulum with inextensible string, you will find that virtual displacements and velocities are the same as the allowed ones, and the last term you'r asking about vanishes.
for another case, think about the same pendulum but with extensible string, say $l = 0.2 t$ .
"the virtual displacement is not always the allowed one, the same for the virtual velocity"
I hope my answer helps you and I think you'll find "Greenwood- Classical Dynamics" useful for you.