I'm trying to identify a distribution presented to me as the "Van Loon distribution".

This is the logistic distribution. \begin{align} \int_0^\delta \frac {\alpha e^{\alpha\eta}}{(1+ e^{\alpha\eta})^2} \,d\eta = \int_2^{1+e^{\alpha\delta}} \frac{du}{u^2} = \frac 1 2 - \frac 1 {1+e^{\alpha\delta}} \to \frac 1 2 \text{ as } \delta\to+\infty. \end{align} Since the density is an even function, as may be checked with a bit of algebra, we have $$ p=\int_{-\infty}^\delta \psi(\eta)\,d\eta = 1 - \frac 1 {1+e^{\alpha\delta}} = \frac 1 {1 + e^{-\alpha\delta}} = \text{a logistic function of }\delta. $$ From this it follows that $$ \delta = \frac 1 \alpha \log \frac p {1-p} = \frac 1 \alpha \operatorname{logit} p. $$ "Logit" is conventionally pronounced with a "long o" as in "boat" and a soft "g" sounding like the "j" in "jet", and the stress on the first syllable.

Google the terms "logistic function" and "logistic distribution."

This seems to be the logistic distribution with location $\mu=0$ and scale $s = 1/\alpha$. As noted by Michael Hardy in the comments, the PDF is symmetric [about $\mu$].