My question is whether the distribution on $\Bbb R$ with probability density $$ f(x) := \frac 2 {\sqrt{2\pi}} e^{-\frac{x^2}{2}} - 2 \vert x\vert \int_{\vert x \vert}^\infty \frac 1{\sqrt{2\pi}} e^{-\frac{y^2} 2} \text d y$$ is already appearing in some context or even has a name or is part of a wider class of distributions.
Here it is. I discovered the following. Let $U_n$ be uniformly distributed on $\{1, \ldots , n \}$ and $X_{n,k}$ normal distributed with mean $0$ and variance $k/n$, independent. Then $$X_{n, U_n} \to Z$$ where $Z$ has the density above. Proof: The characteristic function of $X_{n, U_n}$ converges pointwise to $$t \mapsto \frac{1 - e^{- \frac 1 2 t^2 }}{\frac 1 2 t^2 }$$ By Levy's continuity theorem the laws of $X_{n, U_n}$ have a weak limit. By Fourier inversion one can derive the density of $Z$.