Position in generalized coordinates

Question

In Lagrangian mechanics, when talking about a particle position expressed in generalized coordinates it is usual to find the expression:

$$\mathbf{r}(q_0,...,q_k,t)\tag{1}$$

what it means this isolated $t$ time variable?

Wikipedia uses the expression (see here):

$$\mathbf{r}(\mathbf{q}(t))\tag{2}$$

I can understand all these expressions as equivalent:

$\mathbf{r}(t) = \mathbf{r}(\mathbf{q}(t)) = \mathbf{r}((q_0,...,q_k)(t)) = \mathbf{r}(q_0(t),...,q_k(t)) = \mathbf{r}(q_0,...,q_k)$

being the last one a typographic simplification. But not equivalent to $\mathbf{r}(q_0,...,q_k,t)$

Note we are not talking about a time-dependent vector field defined in a space like in $\mathbf{r}(x,y,z,t)$, but about the coordinates of particles.

Usual expressions from $\mathbf{r}(q_0,...,q_k,t)$ use chain rule including $t$ as an independent variable, like in: $$d\mathbf{r} = \sum_k \frac{\partial \mathbf{r}}{\partial q_k} dq_k + \frac{\partial \mathbf{r}}{\partial t} dt$$.

Answer 1

TL;DR: Yes, OP's eq. (1) is correct and eq. (2) from Wikipedia (July 2020) is wrong/not general enough.

The position

$${\bf r}_i(q^1,\ldots,q^n,t)~\in~ \mathbb{R}^3$$

of the $i$'th point particle, $i\in\{1,\ldots,N\}$, can depend on $n$ generalized coordinates $q^1,\ldots,q^n,$ and explicitly (as well as implicitly) on time $t$. In other words, we assume $3N-n$ holonomic constraints.

For examples of explicit time dependence, see my Phys.SE answers here & here.