From "geometria differenziale" (Abate & Tovena):
$\begin{equation} \text{div}(X)=C_{1}^{1}\nabla X=\partial_iX^i+\Gamma_{i,j}^iX^j \end{equation}$
where Einstein's notation is being used in the local frame expression, and this makes sense to me, but when I try to use this equation in $\mathbb{R^3}$ with spherical coordinates, I get:
$\begin{equation} \text{div}(X)=\partial_rX^r+\partial_\theta X^{\theta}+\partial_\phi X^\phi+\frac{1}{r}X^r+\cot\phi X^\phi+\frac{1}{r}X^r=\\\partial_rX^r+\partial_\theta X^{\theta}+\partial_\phi X^\phi+\frac{2}{r}X^r+\cot\phi X^\phi \end{equation}$
This equation doesn't match what I've found on wikipedia or wolfram:
$\begin{equation} \text{div}(X)=\partial_r X^r+\frac{2X^r}{r}+\frac{\partial_\theta X^\theta}{r\sin\phi}+\frac{\partial_\theta X^\theta}{r}+\frac{\cot\phi}{r}X^{\phi} \end{equation}$
Is this for some normalization?
This page, and I'm assuming whatever Wikipedia page you're talking about, use the convention common in vector calculus where vector fields are written in terms of the orthonormal basis $\hat r,\hat \theta, \hat \phi$ obtained from the coordinate basis by normalizing. Thus, for example, their $\frac{1}{r \sin \phi} \partial_\theta X^\theta$ is equal to your $\partial_\theta X^\theta$ because $\partial_\theta = r \sin \phi\; \hat \theta.$