  By the work of Sela, for any free group $F$, the Coxeter group $W_ 3 =
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$
is elementarily equivalent to $W_3 \ast F$, and so Coxeter groups are not
closed under elementary equivalence among finitely generated groups. In this
paper we show that if we restrict to models which are generated by finitely
many torsion elements (finitely torsion-generated), then we can recover
striking rigidity results. Our main result is that if $(W, S)$ is a Coxeter
system whose irreducible components are either spherical, or affine or (Gromov)
hyperbolic, and $G$ is finitely torsion-generated and elementarily equivalent
to $W$, then $G$ is itself a Coxeter group. This combines results of the second
author et al. from [MPS22, PS23] with the following main hyperbolic result: if
$W$ is a Coxeter hyperbolic group and $G$ is $\mathrm{AE}$-equivalent to $W$
and finitely torsion-generated, then $G$ belongs to a finite collection of
Coxeter groups (modulo isomorphism). Furthermore, we show that there are two
hyperbolic Coxeter groups $W$ and $W'$ which are non-isomorphic but
$\mathrm{AE}$-equivalent. We also show that, on other hand, if we restrict to
certain specific classes of Coxeter groups then we can recover the strongest
possible form of first-order rigidity, which we call first-order
torsion-rigidity, namely the Coxeter group $W$ is the only finitely
torsion-generated model of its theory. Crucially, we show that this form of
rigidity holds for the following classes of Coxeter groups: even hyperbolic
Coxeter groups and free products of one-ended or finite hyperbolic Coxeter
groups. We conjecture that the same kind of phenomena occur for the whole class
of Coxeter groups. In this direction, we prove that if $W$ and $W'$ are even
Coxeter groups which are elementarily equivalent, then they are isomorphic.
