Sure, feel free to include it. Incidentally, it seems to be slightly simpler if you use the generating function $(1-t)/(1-te^{(1-t)x})$, which multiplies all the Eulerian polynomials except the 0th by $t$. This gives just $x/(e^x-1)$ on the right side, though I didn't check to be sure that it's really the same identity.

I don't know of a reference, but I can give you a proof. Would that help?

The question is "I am still wondering if anyone has seen this identity before?" I am saying, yes, this identity (or an equivalent identity) was seen by Tagiuri in 1900.

$$s_\epsilon(1,q,q^2,\dots) = \frac{f^\epsilon(q)}{(1-q)(1-q^2)\cdots (1-q^n)}$$

@MaxAlekseyev: No, zeroing the diagonal doesn't make much difference. It just changes the $2^{i(i+1)/2}$ in $\sum_{i=0}^n s(n,i) 2^{i(i+1)/2}$ to $2^{i(i-1)/2}$.

A closely related problem is that of counting graphs in which no two vertices have the same neighborhood. This corresponds to symmetric 0-1 matrices with 0s on the diagonal. These graphs are called point-determining or mating graphs. You can find links to their enumeration (both labeled and unlabeled) at oeis.org/A006024.

The article linked to is not a eulogy, and at least according to Wikipedia and Khovanskii's web page, he is still teaching at the University of Toronto. His web page is at math.toronto.edu/askold. Why don't you write to him and ask him your question?

I used to have a copy somewhere in my office, but I'm not sure that I can find it. You might try getting it by interlibrary loan if you really want it. The reflection proof can also be found in Krattenthaler's paper referenced below Theorem 10.18.6 in a much more general form.

The reflection principle can be used to find the general formula for Dyck paths of height at most $k$ with no restriction on $k$ referred to in my answer below with the group of reflections isomorphic to the infinite dihedral group. This was first done by Howard D. Grossman, Fun with lattice points, Scripta Math. 15 (1949), 79–81.