Consider the two dimensional sphere $\mathbb{S}^2$ and let $p, q \in \mathbb{S}^2$. Let $\text{exp}_{p}$ and $\text{exp}_{q}$ be the exponential maps on $\mathbb{S}^2$ at points $p$ and $q$ respectively. I am interested in the map $\psi := \text{exp}_{p}^{-1} \circ \text {exp}_{q}$ defined on the unit disc $\mathbb{D} \subset \mathbb{R}^2$. I expect that if $p$ and $q$ are *nearby* points, then the map $\psi$ is close to the identity map. My question is: Is there a way to quantify this closeness? More precisely, is it possible to write the Taylor series expansion of $\psi = (\psi_1, \psi_2)$ around the origin as
\begin{align*}
\psi_1(x,y) &= \psi_1(0) + \frac{\partial \psi_1}{\partial x}(0)~x + \frac{\partial \psi_1}{\partial y}(0)~ y + \ldots \\
\psi_2(x,y) &= \psi_2(0) + \frac{\partial \psi_2}{\partial x}(0)~x + \frac{\partial \psi_2}{\partial y}(0)~ y + \ldots
\end{align*}
and show that the only significant coefficients in the above expansion are $\frac{\partial \psi_1}{\partial x}(0) \approx 1$, $\frac{\partial \psi_2}{\partial y}(0) \approx 1$ and all other partial derivatives $\frac{\partial^{m+n} \psi_i}{\partial^{m} x \partial^{n} y}(0) \approx 0$, for all other choices of $m,n$ and any $i = 1,2$.

Edit: After reading the comments/ answers of user35593 and Gro-Tsen , I realized that the following comments are essential regarding the above question:

First is regrding the comment of user35593 (please read his/her comment below): The question makes sense only when we specify a certain way of identifying the tangent spaces at the points $p$ and $q$. Consider the rotation $R :\mathbb{R}^3 \rightarrow \mathbb{R}^3$ which maps $p$ to $q$, then it also maps $T_{p}(\mathbb{S}^2)$ to $T_{q}(\mathbb{S}^2)$ and it gives a way of identifying the two tangent spaces. The above question I asked is for the tangent spaces identified in this manner.

Second is regarding the answer of Gro-Tsen : One possible interpretation of ``$\frac{\partial^{m+n} \psi_i}{\partial^{m} x \partial^{n} y}(0) \approx 0$'' which I am interested in is: Given $\epsilon >0$, is it possible to find $\delta >0$ such that whenever $d_{\mathbb{S}^2}(p,q)\leq \delta$ we have \begin{align*} \left| \frac{\partial^{m+n} \psi_i}{\partial^{m} x \partial^{n} y}(0) \right| \leq \epsilon \end{align*} whenever $m+n \leq 100$ (except for the cases of $\frac{\partial \psi_1}{\partial x} (0)$ and $\frac{\partial \psi_2}{\partial y} (0)$ which I expect to be $\approx 1$).

Thanks!