Let $(M,g)$ be a Riemannian manifold of dimension $N$ with (Levi-Civita) connection $\nabla$.
I have seen the following definition of Christoffel symbols: For a given smooth moving frame $A=(A_1,\dots, A_N)$ (i.e. $A(p)$ is a basis for the tangent space $TM_p$ at every point $p\in U\subset M$ where $U$ is open), the Christoffel symbols are (locally) defined as the unique $\mathcal C^\infty(U)$-functions $\Gamma_{n,m}^k$ such that $$\nabla_{A_n} A_m =\Gamma_{n,m}^k A_k$$ for $1\le m,n\le N$, where I am using the Einstein convention.
I have seen the following definition for sub-manifolds $M\subset \mathbb R^a$, $a\in\mathbb N$: They are defined (locally) as the unique smooth functions $\Gamma_{n,m}^k$ such that $$\left(\frac{\partial^2 f}{\partial x^n x^m}\right)^\top=\Gamma_{n,m}^k \frac{\partial f}{\partial x^k}$$ for all $1\le n,m\le N$, where $f:\Omega\to M$ is a local parametrization of $M$ with an open set $\Omega\subset\mathbb R^N$ and $^\top$ denotes the tangential projection onto the tangent space of $M$.
Why are these two definitions compatible?
My attempt: Let $M\subset\mathbb R^a$ be an $N$-dimensional smooth manifold that inherits the Riemannian metric from $\mathbb R^a$ by restriction. Let $f=(f^1,f^2,\dots,f^N)$ be a local parametrization around $p\in M$. Take the local frame $A_k=\frac{\partial f}{\partial x^k}$ and let $\nabla^\top$ be the connection that $M$ inherits from $\mathbb R^a$, i.e. if $X$ and $Y$ are two tangential vector fields to $M$, then $\nabla^\top_X Y\overset{\text{Def.}}=(\nabla_{\widetilde X}\widetilde Y)|_M^\top$, where $\widetilde X$ and $\widetilde Y$ are any smooth extensions of $X,Y$ to an open subset of $\mathbb R^n$, and $\nabla$ is defined by $$\nabla_v \widetilde Y = v^j\frac{\partial \widetilde Y^i}{\partial x^j} \left.\frac{\partial}{\partial x^i}\right|_p\in T\mathbb R^a_p$$ for any tangent vector $v\in T\mathbb R^a_p$. I want to show that $\nabla_{A_n} A_m=\left(\frac{\partial^2 f}{\partial x^n\partial x^m}\right)^\top$. For this, fix $n,m\in{1,\dots, N}$ and let $X=A_n=\frac{\partial f}{\partial x^n}, Y=A_m=\frac{\partial f}{\partial x^m}$. I computed:
\begin{split} X_p(Y^i)=\frac{\partial f^j}{\partial x^n}(p)\cdot \frac{\partial\left(\frac{\partial f^i}{\partial x^m}\right)}{\partial x^j}(p)=\frac{\partial f^j}{\partial x^n}(p)\cdot\frac{\partial^2 f^i}{\partial x^m\partial x^j}(p). \end{split}
But shouldn't I get $$X_p(Y^i)=\frac{\partial^2 f^i}{\partial x^n\partial x^m}(p)$$ for the definitions to work out? Where did I go wrong?
Recall that in your formula, you're writing $v=\sum v^j A_j$ (with $A_j = \partial f/\partial x^j$), so in the case of the vector field $v=X=\partial f/\partial x^n$, you will have $v^j = \delta^j_n$.