  We provide a complete description of the presentations of the interval groups
related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced
cases, we show that each interval group is the quotient of the Artin group
associated with the corresponding Carter diagram by the normal closure of a set
of twisted cycle commutators, one for each 4-cycle of the diagram. Our
techniques also reprove an analogous result for the Artin groups of finite
Coxeter groups, which are interval groups corresponding to Coxeter elements. We
also analyse the situation in the non-simply laced cases, where a new Garside
structure is discovered.
  Furthermore, we obtain a complete classification of whether the interval
group we consider is isomorphic or not to the related Artin group. Indeed,
using methods of Tits, we prove that the interval groups of proper
quasi-Coxeter elements are not isomorphic to the Artin groups of the same type,
in the case of $D_n$ when $n$ is even or in any of the exceptional cases. In
[BHNR22], we show using different methods that this result holds for type $D_n$
for all $n \geq 4$.
