Heat equation of the distance function and Ricci flow

Question

I would like to get more acquainted to with Ricci flow and evolving metrics. Suppose we are given a familly $\{M, g(t)\}_t$ of Riemannian manifolds with time-dependent metrics. We would like to understand what evolution on the distance function could be set in order to get Ricci flow.

1. Suppose that the distance function evolves according to the following modified heat equation: $$ \partial_t d^2_{g(t)}\left(x, y\right) = \frac{1}{2}\Delta^{(t)}_2 d^2_{g(t)}\left(x, y\right) - n, \quad \forall x, y \in M.$$ Here $d^2_{g(t)}$ denotes the distance square w.r.t. to $g(t)$ and $\Delta^{(t)}_2$ the Laplace-Beltrami operator w.r.t. the 2nd variable and the metric at time $t$, and $n$ stands for the dimension of the manifold $M$.
2. The evolution of the distance function defines a family of metrics such that this equation holds. What can be said about the metric evolution ?

Here is an attempt. $$ \partial_t g(t; x)[v, v] = \partial_t \lim_{s\to 0} \frac{d^2_{g(t)}\left(x, \gamma(s)\right)}{s^2}, $$ with $\gamma$ being any curve such that $\gamma(0) = x$ and $\gamma'(0) = v$. Now, by interchanging limits, we deduce \begin{align} \partial_t g(t; x)[v, v] &= \lim_{s\to 0} \frac{1}{s^2} \partial_t d^2_{g(t)}\left(x, \gamma(s)\right) \\ &= \lim_{s\to 0} \frac{1}{s^2}\left(\frac{1}{2}\Delta^{(t)}_2 d^2_{g(t)}\left(x, \gamma(s)\right) - n\right). \end{align} But since this holds for any $\gamma$, and since this is a partial (and not total) derivative, we can use a time-dependent $\gamma$ as well. We will use $\gamma^{(t)}(s)$ to be the geodesic with initial velocity $v$ at time $s$ w.r.t. to the metric $g(t)$. Using the expansion of the laplacian in spherical coordinates (see also), we get \begin{align} \partial_t g(t; x)[v, v] &= \lim_{s\to 0} \frac{1}{s^2} \left( \Delta^{(t)}_2 d^2_{g(t)}\left(x, \exp_x^{(t)}(sv)\right) - n\right)\\ &=\lim_{s\to 0} \frac{1}{s^2} \left(n - \frac{1}{3}\mathrm{Ric}(t;x)[v, v]s^2 + O(s^3) - n\right) \\ &= - \frac{1}{3}\mathrm{Ric}(t;x)[v, v]. \end{align} It seems that the metric is evolving according to the Ricci flow. Does this make sense ? I can not find any reference mentionning this fact. Otherwise, what is wrong in essence?