I am learning about connections using Lee's Intro to Riemannian Manifolds, and I am confused regarding the coefficients of the pullback connection. For the setting, let $(M,g) $ be a Riemannian manifold and let $\nabla$ be a connection on $M$. From Prop 4.7 in Lee's book, if we have two local frames $(E_i)$ and $(\tilde{E}_j)$ related by a matrix $A^j_i$, then the transformation law for the connection coefficients is: $$ \tilde{\Gamma}^k_{ij} = (A^{-1})^k_pA^q_iA^r_j\Gamma^p_{qr} + (A^{-1})^k_pA^q_iE_q(A^p_j). \tag{1} $$
I showed this as indicated in this question, using the definition of the connection coefficients. Further, Lee introduces the pullback connection by a diffeomorphism $\varphi:M \to \tilde{M}$ between $(M,g) $ and $(\tilde{M},\tilde{g})$ by: $$ (\varphi^*\tilde{\nabla})_X Y = (\varphi^{-1})_*(\tilde{\nabla}_{\varphi_*X}(\varphi_*Y)), $$ for any vector fields $X,Y$ on $M$. I obtained that the coefficients of the pullback connection satisfy an analog relation to the transformation law above. If $x^i$ are coordinates on $M$ and $\tilde{x}^j$ are coordinates on $\tilde{M}$ and denoting the coefficients of $\varphi^*\tilde{\nabla}$ by ${}^{\varphi^*\tilde{\nabla}}\Gamma^k_{ij}$ and those of $\tilde{\nabla}$ by ${}^{\tilde{\nabla}}\Gamma^k_{ij}$, I obtained: $$ {}^{\varphi^*\tilde{\nabla}}\Gamma^k_{ij} = \frac{\partial \varphi^{-1,k}}{\partial \tilde{x}^p} \frac{\partial \varphi^q}{\partial x^i} \frac{\partial \varphi^r}{\partial x^j}{}^{\tilde{\nabla}}\Gamma^p_{qr} + \frac{\partial \varphi^{-1,k}}{\partial \tilde{x}^p} \frac{\partial \varphi^q}{\partial x^i} \frac{\partial}{\partial \tilde{x}^q}\left( \frac{\partial \varphi^p}{\partial x^j} \right).\tag{2} $$
Now, according to this question and the follow-up discussion here, the coefficients of the pullback connection take the simpler form (using my notation): $$ {}^{\varphi^*\tilde{\nabla}}\Gamma^k_{ij} = {}^{\tilde{\nabla}}\Gamma^k_{im}(d\varphi)^m_j.\tag{3} $$ As I understand it, $d\varphi$ is the differential of $\varphi$, so in components it simply writes $\frac{\partial \varphi^m}{\partial x^j}$, and we are far from (2).
Question 1:Does each expression for ${}^{\varphi^*\tilde{\nabla}}\Gamma^k_{ij} $ apply to a particular case? Which is the correct one? (This will probably help me understand better the situation in my other question regarding developable surfaces)
Question 2: More general question: (1) and (2) are identical when $\varphi$ is a change of variable from $M$ to $M$. Is it correct to view a change of variable as a pullback? Do the pullback relations always reduce to the relevant change of variable formulas when the map $\varphi$ is a change of variable? For instance, the components of the pullback of a 2-covariant tensor match the components of the tensor after a change of variable.
Thanks a lot for the help.
Edit: I obtained (2) by applying the pointwise pullback on basis vectors, with $F = \varphi$: \begin{align} (F^*\tilde{\nabla})_{\frac{\partial}{\partial x^i}\Large\vert_{p}}\left( \frac{\partial}{\partial x^j}\right) &= dF^{-1}_{F(p)}\left[ \tilde{\nabla}_{dF_p\left(\frac{\partial}{\partial x^i}\large\vert_{p}\right)} \left( dF_p\left( \frac{\partial}{\partial x^j}\right)\right)\right]\\ &= dF^{-1}_{F(p)}\left[ \tilde{\nabla}_{\frac{\partial F^k}{\partial x^i}\Large\vert_{p}\frac{\partial}{\partial \tilde{x}^k}} \left( \frac{\partial F^l}{\partial x^j}\biggr\vert_{p}\frac{\partial }{\partial \tilde{x}^l}\right)\right]\\ &= dF^{-1}_{F(p)}\left[ \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \tilde{\nabla}_{\frac{\partial}{\partial \tilde{x}^k}} \left( \frac{\partial F^l}{\partial x^j}\biggr\vert_{p}\frac{\partial }{\partial \tilde{x}^l}\right) \right]\\ &= dF^{-1}_{F(p)}\left[ \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \left( \frac{\partial}{\partial \tilde{x}^k}\biggr\vert_{F(p)} \left( \frac{\partial F^l}{\partial x^j}\right) \frac{\partial }{\partial \tilde{x}^l} + \frac{\partial F^l}{\partial x^j}\biggr\vert_{p} \tilde{\nabla}_{\frac{\partial}{\partial \tilde{x}^k}} \left( \frac{\partial }{\partial \tilde{x}^l}\right) \right)\right]\\ &= dF^{-1}_{F(p)}\left[ \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \left( \frac{\partial}{\partial \tilde{x}^k}\biggr\vert_{F(p)} \left( \frac{\partial F^l}{\partial x^j}\right) \frac{\partial }{\partial \tilde{x}^l} + \frac{\partial F^l}{\partial x^j}\biggr\vert_{p} {}^{\tilde{\nabla}}\Gamma^m_{kl} \frac{\partial}{\partial \tilde{x}^m}\right)\right]\\ &= \left[ \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \left( \frac{\partial}{\partial \tilde{x}^k}\biggr\vert_{F(p)} \left( \frac{\partial F^m}{\partial x^j}\right) + \frac{\partial F^l}{\partial x^j}\biggr\vert_{p} {}^{\tilde{\nabla}}\Gamma^m_{kl} \right)\right] dF^{-1}_{F(p)} \left(\frac{\partial}{\partial \tilde{x}^m} \right)\\ &= \left[ \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \left( \frac{\partial}{\partial \tilde{x}^k}\biggr\vert_{F(p)} \left( \frac{\partial F^m}{\partial x^j}\right) + \frac{\partial F^l}{\partial x^j}\biggr\vert_{p} {}^{\tilde{\nabla}}\Gamma^m_{kl} \right)\right] \frac{\partial F^{-1,r}}{\partial \tilde{x}^m} \frac{\partial}{\partial x^r} \\ &= \left( \frac{\partial F^{-1,r}}{\partial \tilde{x}^m}\biggr\vert_{F(p)} \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \frac{\partial}{\partial \tilde{x}^k}\biggr\vert_{F(p)} \left( \frac{\partial F^m}{\partial x^j}\right) + \frac{\partial F^{-1,r}}{\partial \tilde{x}^m}\biggr\vert_{F(p)} \frac{\partial F^k}{\partial x^i}\biggr\vert_{p} \frac{\partial F^l}{\partial x^j}\biggr\vert_{p} {}^{\tilde{\nabla}}\Gamma^m_{kl} \right) \frac{\partial}{\partial x^r} \\ &= \left( \frac{\partial F^{-1,k}}{\partial \tilde{x}^p}\biggr\vert_{F(p)} \frac{\partial F^q}{\partial x^i}\biggr\vert_{p} \frac{\partial}{\partial \tilde{x}^q}\biggr\vert_{F(p)} \left( \frac{\partial F^p}{\partial x^j}\right) + \frac{\partial F^{-1,k}}{\partial \tilde{x}^p}\biggr\vert_{F(p)} \frac{\partial F^q}{\partial x^i}\biggr\vert_{p} \frac{\partial F^r}{\partial x^j}\biggr\vert_{p} {}^{\tilde{\nabla}}\Gamma^p_{qr} \right) \frac{\partial}{\partial x^k} \\ &= {}^{F^*\tilde{\nabla}}\Gamma^k_{ij} \frac{\partial}{\partial x^k} \\ \end{align}
You misunderstood the other post. The two formulisms are compatible, although you can say the other post has much less information.
First, the other post is using a different convention of the Christoffel symbols -- he differentiates the first vector field with respect to the second. So your $\Gamma_{ij}^k$ is his $\Gamma_{ji}^k$. Then his formula, as you correctly tried to interpret, is $$ {}^{\varphi^*\tilde\nabla}\Gamma_{ij}^k =\frac{\partial \varphi^q}{\partial x^i} {}^{\tilde\nabla}\Gamma_{qj}^k . $$ This would match with your formula (2) if you group terms as $$ {}^{\varphi^*\tilde\nabla}\Gamma_{ij}^k =\frac{\partial \varphi^q}{\partial x^i}\Big(\frac{\partial\varphi^{-1,k}}{\partial \tilde x^p} \frac{\partial\varphi^{r}}{\partial x^j}{}^{\tilde\nabla}\Gamma_{qr}^p + \frac{\partial\varphi^{-1,k}}{\partial \tilde x^p} \frac{\partial}{\partial \tilde x^q}\big(\frac{\partial\varphi^{p}}{\partial x^j}\big) \Big). $$ So the compatibility comes from $$ {}^{\tilde\nabla}\Gamma_{qj}^k=\frac{\partial\varphi^{-1,k}}{\partial \tilde x^p} \frac{\partial\varphi^{r}}{\partial x^j}{}^{\tilde\nabla}\Gamma_{qr}^p + \frac{\partial\varphi^{-1,k}}{\partial \tilde x^p} \frac{\partial}{\partial \tilde x^q}\big(\frac{\partial\varphi^{p}}{\partial x^j}\big). $$
The thing to note is that $i, j, k$ are indices on $M$ for $x^i, x^j, x^k$, and $p, q, r$ are indices on $\tilde M$ for $\tilde x^p, \tilde x^q, \tilde x^r$. So the notation $$ {}^{\tilde\nabla}\Gamma_{qj}^k $$ is not very well-defined, since it involves both sets of variables. Your formula would be a correct working out of it by $$ \varphi_*\frac{\partial}{\partial x^j} = \frac{\partial \varphi^r}{\partial x^j} \frac{\partial}{\partial \tilde x^r},\quad (\varphi^{-1})_*\frac{\partial}{\partial \tilde x^p} = \frac{\partial \varphi^{-1,k}}{\partial \tilde x^p} \frac{\partial}{\partial x^k}. $$ I believe you can fill in this part.
From this point of view, the other post is only saying that $\Gamma_{ij}^k$ is tensorial in the index $i$, since that is the index for $X$ in $\nabla_X Y$.
Now about your Question 2, it is not a coincidence that formula (1) and (2) are identical. A change of variable is a diffeomorphism on the coordinate neighborhood. So a pullback by a diffeomorphism and a change of variables are about the same, at least locally.