Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ideal sheaf of $x$ then we can define the blow-up to be $\text{Proj} \ \mathcal{O}_X \oplus \mathcal{I}_x \oplus \mathcal{I}_x^2 \oplus \cdots$. The other, which works only when $X$ is quasi-projective, is geometric but not intrinsic: one chooses an embedding $X \to \mathbb{P}^n$, defines the blow-up of $\mathbb{P}^n$ at a point explicitly in coordinates, and then takes the proper transform (or whatever the terminology is) of $X$.
I hope there is a construction which is both intrinsic and geometric (it may be nothing but a repackaging of the $\text{Proj}$ definition). Here is a starting point to indicate what I'm looking for. Suppose $x \in X$ is a nonsingular point for simplicity, so the tangent space $T_xX$ is well-behaved. As I understand it, the blow-up is set-theoretically $X \setminus \{ x \} \sqcup \mathbb{P}(T_xX)$, but obviously this is not a disjoint union in the sense of varieties. How does one make this into a variety?