In Lagrangian mechanics, the Lagrangian $L=L(\mathbf{q},\mathbf{\dot q},t)$ is a real-valued function from the product $TM \times \mathbb{R}$ between the tangent bundle $TM$ of the configuration space $M$ and the time axis $\mathbb{R}$. Since $TM \times \mathbb{R}$ is a manifold, $L$ must not depend on a specific coordinate representation. This, in turn, means that $L$ must be invariant under the transformation $$ (\mathbf{q}, \mathbf{\dot q}, t) = F(\mathbf{s},\mathbf{\dot s},\tau), $$ where $F \in C^\infty(\mathbb{R}^{2n+1},\mathbb{R}^{2n+1})$ is invertible.
Wikipedia, on the other hand, only claims that $L$ is invariant under the transformation $$ (\mathbf{q},\mathbf{\dot q},t) = (\mathbf{q}(\mathbf{s},\tau),\mathbf{\dot q}(\mathbf{s},\mathbf{\dot s},\tau),\tau) $$ which is much more restrictive. So, who is right?
If Wikipedia is right, my second (closely related) question is: can $\mathbf{\dot q}(\mathbf{s},\mathbf{\dot s},t)$ be any smooth function, or it must be $$\dot {q}^i = \sum_j \frac{\partial q^i}{\partial s^j}\,{\dot s}^j?$$
Both are right, but your transformations are not (necessarily) bundle maps. Then is stands to reason if the word "invariant" applies, what of $L$ stays invariant under such a general map?
The charts of the tangent bundle are usually directly obtained from the charts of the manifold, if you change the chart of the manifold, the change in the tangent bundle map is automatic, and as the implied tangent bundle charts are linear in the direction of the tangent space, the map between the charts will be linear in the same way. This is what the wikipedia article uses.