A well known formula for interpreting Ricci curvature is the following (see for example Wikipedia's article). Consider the Taylor expansion of the volume form at a point $p$ in normal coordinates $\left(x_i\right)$: $$ d\mu = 1- \frac{1}{6}\mathrm{Ric}_{ij}x^ix^j + O(|x|^3)d\mu_\text{Euclidean}.$$ In particular, this gives an interpretation for $\mathrm{Ric}_{ii}$ in terms of volume of "infinitesimal cones starting at $p$ in the direction $x_i$".
On the other hand, it is well known that in dimension 2 the Ricci curvature is just $\mathrm{Ric}=\frac{\mathrm{R}}{2}g$. This means that (in normal coordinates) $\mathrm{Ric}_{11}=\mathrm{Ric}_{22}$.
My question is, how can I reconcile these two points of view? Why do these "infinitesimal cones" behave the same in two dimensions (but not in higher dimensions)?