The genuinely multipartite nonlocality of graph states is model-dependent.

The notion of genuinely multipartite nonlocality (GMNL) was introduced to conceptualize the fact that nonclassical resources involving more than two parties in a nontrivial way may be needed to account for some quantum correlations. In this letter, we first recall the contradictions inherent to the historical definition of GMNL. Second, we turn to one of its redefinitions, called Local-Operations-and-Shared-Randomness GMNL (LOSR-GMNL), proving that all caterpillar graph states (including linear cluster states) have this second property. Finally, we conceptualize a third, alternative definition, which we call Local-Operations-and-Neighbour-Communication GMNL (LONC-GMNL), that is adapted to situations in which short-range communication between some parties might occur. We show that linear cluster states do not have this third property, while GHZ states do. Beyond its technical content, our letter illustrates that rigorous conceptual work is needed before applying the concepts of multipartite nonlocality or entanglement to benchmark the nonclassicality of some experimentally-produced quantum systems.