Definition. Let $M$ be a smooth manifold. Then a distribution on $M$ of rank $k$ is defined to be a (usually smooth) rank-$k$ subbundle of the tangent bundle $TM$.[1]
Definition. Let $X$ be an affine scheme over $k$, and let $x$ be a $k$-valued point. Write $I_x$ for the set of functions $f \in \mathcal{O}(X)$ such that $f(x) = 0$. A distribution on $X$ with support in $x$ of order $\leq n$ is a linear functional $\mu : \mathcal{O}(X) \to k$ such that $\mu(I_x^{n+1}) = 0$.[2]
I can only assume that the second definition is based on the first, but I do not see where the definition comes from. Why is the definition of a distribution on an affine scheme the way it is?
[1]: Lee, John M., Introduction to smooth manifolds, Graduate Texts in Mathematics. 218. New York, NY: Springer. xvii, 628 p. (2002). ZBL1030.53001.
[2]: Jantzen, Jens Carsten, Representations of algebraic groups, Pure and Applied Mathematics, Vol. 131. Boston etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XIII, 443 p.; {$} 59.50 (1987). ZBL0654.20039.
This is a big old confusion regarding the two different notions of distributions floating around in mathematics. For one, there are (geometrical) distributions as subbundles of tangent bundles, as in your first definition. On the other hand, there is the notion of (analytical) distributions as functionals on function spaces, for example the Dirac delta functionals and its derivatives, so $$\delta_p : C^\infty(\mathbb{R}) \to \mathbb{R}, f \mapsto f(p)$$ or $$\delta_p' : C^\infty(\mathbb{R}) \to \mathbb{R}, f \mapsto f'(p)$$ for $p \in \mathbb{R}$. This is what your second definition generalizes, to affine schemes specifically. The order in your definition then mimics the idea of the order of a differential operator, but in particular, this has nothing to do with the rank of the geometrical distributions.
As far as I know, there is no direct relation in any way. It would be pretty helpful if people added the adjectives "geometrical" or "analytical", but I'm sure someone in the math community will take issue with that, so for now, whenever you come across the word "distribution", be very careful :)