Considering a topological manifold $\mathcal S$, let's say specificly of ${\rm dim} = 2$ (i.e. "a surface") we can also identify curves (the set of curves $\{ \mathcal K_j \}$) "in" this surface, i.e. $\mathcal K_j \subset \mathcal S$.
For any point $p \in \mathcal S$ we can distinguish curves "through" this point (i.e. set $\{ \mathcal K_j : p \in K_j \} \equiv \{ \mathcal K^p_j \}$ from all other curves.
(Note that this set $\{ \mathcal K^p_j \}$ of curves through point $p$ generally contains pairs of curves $\mathcal K^p_a, \mathcal K^p_b \in \{ \mathcal K^p_j \}$ for which there exists a neighbourhood $\mathcal U_p \subset \mathcal S$ of point $p$ such that point $p$ is the only point of this neigborhood which these two curves have in common:
$$((\mathcal K^p_a \cap \mathcal U_p) \cap (\mathcal K^p_b \cap \mathcal U_p)) = p.$$
If so, then, or course, there are also many other neighborhoods with the same property in respect to these two specific curves, $\mathcal K^p_a$ and $\mathcal K^p_b$.)
Now, it may be useful to consider some (non-zero but perhaps not necessarily proper) subset $\Gamma^p \subseteq \{ \mathcal K^p_j \}$ of curves through point $p$, and to partition set $\Gamma^p$ in disjoint equivalence classes, regarding equivalence by (pairwise mutual) tangentiality of curves of each one of such classes of curves "tangent to each other in point $p$". However, the method described in Wikipedia, as well as some variant method described and to be considered below, is not directly applicable to just any surface $\mathcal S$ (given as topological manifold of ${\rm dim} = 2$), but these methods require "a $C^k$ differentable manifold (with smoothness $k \ge 1$)".
As far as I understand, each surface $\mathcal S$ which is characterized as a topological manifold by its topology thereby has a unique maximal $C^0$-atlas $\mathsf A^{(0)}$ of "smoothness" $k = 0$; a.k.a. the maximal continuous atlas of $\mathcal S$. (If this happens to be wrong, then please let me know; and I'd like you to consider in the following just one partcular maximal $C^0$-atlas $\mathsf A^{(0)}$ of the given surface $\mathcal S$.)
So, in order to apply the method mentioned above, as well as its variant, for establishing disjoint equivalence classes of curves "tangent to each other in point $p$" on a suitable set of curves $\Gamma^p$, we should select one specific (not necessarily maximal) $C^1$-atlas $\mathsf A^{(1)}$ of smoothness $k = 1$, as subset of the given maximal $C^0$-atlas $\mathsf A^{(0)}$.
The method under consideration then requires to pick a particular coordinate chart $\varphi_w \in (\mathcal U^p_w, \varphi_w) \in \mathsf A^{(1)}$, where of course $p \in \mathcal U^p_w$, and shall then proceed as follows:
- To each curve $\mathcal K^p_j \in \{ \mathcal K^p_j \}$ assign a corresponding parametrization $\gamma^{\mathsf A}_j : (-1, 1) \leftrightarrow \mathcal K^p_j$ such that $\gamma^{\mathsf A}_j[ \, 0 \, ] := p$ and, if at all possible, such that the coordinate representation of curve $\mathcal K^p_j$ in coordinate chart $\varphi_w$ is differentiable with respect to variable $t \in (-1, 1)$ at least at value $t = 0$, i.e at least at (the image of) point $p$; and such that the value of the derivative at value $t = 0$ is different from $(0, 0) \in \mathbb R^2$ :
$$ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma^{\mathsf A}_j) \, \right]_{(t = 0)} \text{ exists and is } \ne (0,0).$$
Those curves $\mathcal K^p_j$ for which such a suitable parametrization $\gamma_j^{\mathsf A}$ can be found become members of the set $\Gamma^p_{\mathsf A}$ of (images of parametrized) curves through point $p$.
Any two curves $\mathcal K^p_a, \mathcal K^p_m \in \Gamma^p_{\mathsf A}$ are then said to be equivalent (and thus members of the same equivalence class) if and only if for the corresponding parametrized curves (a.k.a. paths) $\gamma^{\mathsf A}_a, \gamma^{\mathsf A}_m$ holds:
$$ \exists \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} = r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_m^{\mathsf A}) \, \right]_{(t = 0)}.$$
Importantly, it is claimed in connection with the original method, as described in the Wikipedia section linked above, that the equivalence classification (partition) of set $\Gamma^p_{\mathsf A}$ which is thereby achieved "does not depend on the choice of coordinate chart" $\varphi_w$. (This appears to be widely appreciated and proven ... especially clearly here or here.)
Surely, the chart-independence considered and proven refers to chart-choices restricted to charts $\{ \varphi_w : \varphi_w \in (\mathcal U^p_w, \varphi_w) \}$ of neighbourhoods $\{ \mathcal U^p_w \}$ which all must contain point $p$ (i.e. restricted to charts which "map point $p$"), and restricted to charts from one-and-the-same (possibly maximal) $C^1$-atlas $\mathsf A^{(1)}$.
However: along with $C^1$-atlas $\mathsf A^{(1)}$, the given maximal $C^0$-atlas $\mathsf A^{(0)}$ can (and, being maximal, surely does) also contain another $C^1$-atlas, $\mathsf B^{(1)}$, which is non-equivalent to atlas $\mathsf A^{(1)}$, and which in particular contains a chart $\psi_y \in (\mathcal V^p_y, \psi_y) \in \mathsf B^{(1)}$ which is incompatible, at point $p$, to any chart $\varphi_w$ of atlas $\mathsf A^{(1)}$ which is mapping point $p$; i.e. such that for the transition function
$$(\psi_y \circ \varphi_w^{(-1)}) : \{ (\text{image of } \varphi_w[ \, \mathcal U^p_w \cap \mathcal V^p_y \, ]) \subset \mathbb R^2 \} \longleftrightarrow \{ (\text{image of } \psi_y[ \, \mathcal V^p_y \cap \mathcal U^p_w \, ]) \subset \mathbb R^2 \}$$
either this transition function $(\psi_y \circ \varphi_w^{(-1)})$ itself fails to be differentiable "at (the image of) point $p$", i.e. at $\varphi_w[ \, p \, ] \in \mathbb R^2$,
or the corresponding inverse transition function, $(\psi_y \circ \varphi_w^{(-1)})^{(-1)} := (\varphi_w \circ \psi_y^{(-1)})$ fails to be differentiable "at (the image of) point $p$", i.e. at $\psi_y[ \, p \, ] \in \mathbb R^2$,
or both.
(The necessary construction or identification of such an atlas $\mathsf B^{(1)}$ and chart $\psi_y$ can be as easy as sketched here.)
The described method variant for establishing equivalence classes of curves "tangent to each other in point $p$" can now similarly be carried out by
picking out chart $\psi_y$,
assigning the required parametrizations, if at all possible, to each curve $\mathcal K^p_j$ with respect to chart $\psi_y$,
collecting the (images of) suitably parametrized curves as set $\Gamma^p_{\mathsf B}$,
partitioning set $\Gamma^p_{\mathsf B}$ into disjoint equivalence classes; explicitly:
Any two curves $\mathcal K^p_b, \mathcal K^p_n \in \Gamma^p_{\mathsf B}$ are said to be equivalent (and thus members of the same equivalence class) if and only if for the corresponding paths $\gamma^{\mathsf B}_b, \gamma^{\mathsf B}_n $ holds:
$$ \exists \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_b^{\mathsf B}) \, \right]_{(t = 0)} = s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_n^{\mathsf B}) \, \right]_{(t = 0)}.$$
My questions:
(1) Given any surface $\mathcal S$ and picking any chart $\varphi_w$ (which maps point $p$) from a suitable $C^1$-atlas $\mathsf A^{(1)}$, can always an inequivalent $C^1$-atlas $\mathsf B^{(1)}$ be found or constructed, and a suitable chart $\psi_y$ be picked from it, such that there exists at least one curve $\mathcal K^p_j$ at all which can be suitably parametrized wrt. chart $\varphi_w$ as well as wrt. chart $\psi_y$ (i.e. such that the image of one corresponding path $\gamma_j^{\mathsf A}$ becomes a member of set $\Gamma^p_{\mathsf A}$, and the image of the other corresponding path $\gamma_j^{\mathsf B}$ becomes a member of set $\Gamma^p_{\mathsf B}$) ?
(2) In case that there are several curves $\mathcal K^p_j$ which can be suitably parametrized wrt. chart $\varphi_w$ as well as wrt. chart $\psi_v$, are the resulting partitions of $\Gamma^p_{\mathsf A}$ and of $\Gamma^p_{\mathsf B}$ guaranteed to be consistent for these curves, and thereby "independent of the choice of charts" even from inequivalent atlases $\mathsf A^{(1)}$ and $\mathsf B^{(1)}$ ?
Or more formally: is guaranteed that
$$ {\mathbf {if}} \, \, \exists \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} = r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_m^{\mathsf A}) \, \right]_{(t = 0)} \\ {\mathbf {then}} \, \, \exists \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_a^{\mathsf B}) \, \right]_{(t = 0)} = s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_m^{\mathsf B}) \, \right]_{(t = 0)},$$
for applicable curves $\mathcal K^p_a$ and $\mathcal K^p_m$, and
$$ {\mathbf {if}} \, \, \forall \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} \ne r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_b^{\mathsf A}) \, \right]_{(t = 0)} \\ {\mathbf {then}} \, \, \forall \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_a^{\mathsf B}) \, \right]_{(t = 0)} \ne s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_b^{\mathsf B}) \, \right]_{(t = 0)}$$
for applicable curves $\mathcal K^p_a$ and $\mathcal K^p_b$ ?
