Does anyone know if there is an intrinsic definition of a toroidal compactification (over $\mathbb{C}$)?
Something like: Let $X$ be an algebraic variety over $\mathbb{C}$. Then $X \subset \bar{X}$ is a toroidal compactification if for every $x \in \bar{X}\setminus X$ there is an analytic neighborhood $N(x)$ of $x$ such that $N(x)$ is isomorphic to a toric variety. If there is, could someone give me any reference for it?
The standard definition of toroidal embedding from the book $\textit{Toroidal Embeddings}$ by Kempf, Knudsen, Mumford, and Saint-Donat is very similar to your proposed definition.
I'll use your notation. We say $X \subset \bar{X}$ where $X$ is smooth and $\bar{X}$ is normal and $n$-dimensional is a toroidal embedding if for every point $x \in \bar{X}$ there exists an affine toric variety $X_{\sigma}$ with $n$-dimensional torus $T$ and an isomorphism of completed local rings
$$ \hat{\mathcal{O}}_{\bar{X},x} \to \hat{\mathcal{O}}_{X_{\sigma},t} $$
for $t \in X_{\sigma}$ such that this isomorphism maps the ideal of the boundary divisor $\bar{X} \setminus X$ to the ideal of the toric boundary divisor $X_\sigma \setminus T$.
This last condition is very important because it is what ensures that the boundary divisor has a stratification that behaves like the stratification into orbits of a toric variety and thus can be described combinatorially by a fan.
Often in the literature people will restrict to a slightly better behaved class called toroidal without self intersection. This just means the above definition and the extra condition that each component of the boundary divisor $\bar{X} \setminus X$ is normal.