$\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$Let $X$ be a smooth variety. The Hilbert scheme of two points $X^{[2]}$ on $X$ can be obtained by blowup the diagonal of the symmetric product $\Sym^2X:=(X\times X)/\mathbb Z_2$: $$X^{[2]}\cong\Bl_{\Delta}(\Sym^2X)\to \Sym^2X.$$

By definition, this space represents the Hilbert functor \begin{gather*} \Hilb^2_X:\Sch^\text{op}\to \Sets \\ T\mapsto \{{\text{flat families of subschemes of $X$ of length 2 over $T$}}\}/{\sim}. \end{gather*} I'm interested in the space $\Bl_{\Delta}(X\times X)$, which is a double cover of $X^{[2]}$ branched along the exceptional divisor. In fact it is obtained as the fiber product square: $\require{AMScd}$ \begin{CD} \Bl_{\Delta}(X\times X) @>>> \Bl_{\Delta}(\Sym^2X)\\ @VVV @VVV\\ X\times X @>>> \Sym^2X. \end{CD}

I'd like to know if there is a modular interpretation of the double cover $\Bl_{\Delta}(X\times X)$. To me, it should be something like the "Hilbert scheme" of ordered two points: Its general point parameterizes two points with an order; When the two points collide and become a fat point, it forget the order. However, I don't know how to formulate the modularity more precisely.

So perhaps here is what I want to ask: Is there a moduli functor $\mathcal{M}:\Sch^\text{op}\to \Sets$ representable by the variety $\Bl_{\Delta}(X\times X)$?