For prime $q \geq 5$ write the count as
$$
\frac1{1152} q (q-1) (q^3 - 21q^2 + 171 q - c_q)
$$
where
$$
c_q = 483 + 36 \left(\frac{-1}{q}\right) + 64 \left(\frac{-3}{q}\right)
+ \delta_q.
$$
Then for $(\frac{-2}{q}) = -1$ **Ronald Bacher**'s calculations
indicate $\delta_q=0$. If $(\frac{-2}{q}) = +1$ then
$q$ can be written as $m^2 + 2n^2$, uniquely up to changing
$(m,n)$ to $(\pm m, \pm n)$, and we have
$$
\delta_q = 24(m^2 - 2n^2) + 192 + 72 \left(\frac{-1}{q}\right).
$$

The explanation is as follows. Start as did **Will Sawin**
by considering the variety of $(s_1,s_2,s_3,s_4,t_1,t_2,t_3,t_4)$
such that for $i=1,2,3$ the $i$-th elementary symmetric function
of the $s$'s equals the $i$-th elem.sym.fn. of the $t$'s.
We may apply any $aX+b$ transformation to all $8$ variables,
which explains the $q(q-1)$ factor. (The factor $1152 = 2 \cdot 4!^2$
is from coordinate permutations that respect the partition of the
$8$ variables into two sets of $4$.) In odd characteristic, there's
a unique representative with $\sum_{i=1}^4 s_i = \sum_{i=1}^4 t_i = 0$;
this takes care of the translations, and then we mod out by scalars
by going to projective space. We end up with the complete intersection
of a quadric and a sextic in ${\bf P}^5$. This threefold, call it
${\cal M}$, turns out to be rational. (This has probably been known
for some time, because ${\cal M}$ classifies perfect multigrades of order $4$,
and such things have been studied since the mid-19th century, see the
Prouhet-Tarry-Escott problem; I outline a proof below.)
However, by requiring that all coordinates be distinct
we're removing some divisor ${\cal D}$ on this threefold,
so the final count decreases by the outcome of an inclusion-exclusion formula
whose terms are point counts over some subvarieties of ${\cal M}$.
Most of these sub varieties are rational curves, or points that may be defined
over ${\bf Q}(i)$ or ${\bf Q}(\sqrt{-3})$, the latter explaining
the appearance of Legendre symbols $(\frac{-1}{p})$, $(\frac{-3}{p})$
in the counting formula. But the two-dimensional components of ${\cal D}$
are isomorphic K3 surfaces, arising as a complete intersection of
a quadric and a cubic in ${\bf P}^4$; and those components make a more complicated
contribution. Fortunately these K3 surfaces are "singular" (i.e. their
Picard number attains the maximum of $20$ for a K3 surface in
characteristic zero) $-$ I computed that they're birational with
the universal elliptic curve over $X_1(8)$ $-$ and it is known that
the point-count of this singular K3 surface can be given by a formula
that involves $m^2-2n^2$ when $(\frac{-2}{q}) = +1$.

To show that $\cal M$ is rational, it is convenient to apply a linear
change of variables from the "$A_3$" coordinates $s_i,t_i$ to
"$D_3$" coordinates, say $a,b,c$ and $d,e,f$, with
$$
s_i = a+b+c, \phantom+ a-b-c, \phantom+ -a+b-c, \phantom+ -a-b+c
$$
and likewise $t_i = d+e+f, \phantom. d-e-f, \phantom. -d+e-f, \phantom. -d-e+f$.
Then $\sum_{i=1}^4 s_i = \sum_{i=1}^4 t_i = 0$ holds automatically,
and the quadric and cubic become simply
$$
a^2+b^2+c^2 = d^2+e^2+f^2,
\phantom\infty
abc = def.
$$
Let $d=pa$ and $e=qb$. Then $f=(pq)^{-1}c$, and the quadric becomes
a conic in the $(a:b:c)$ plane with coefficients depending on $p,q$:
$$
(p^2-1)a^2 + (q^2-1)b^2 + ((pq)^{-2}-1) c^2 = 0.
$$
So $\cal M$ is birational to a conic bundle over the $(p,q)$ plane,
and this conic bundle has a section $(a:b:c:d:e:f) = (1:p:pq:p:pq:1)$
which lets us birationally identify $\cal M$ with the product of the
$(p,q)$ plane with ${\bf P}^1$. This is a rational threefold, **QED**.