Dynamically relaxed galaxy clusters have long played a role in galaxy cluster studies because it is thought their properties can be reconstructed more precisely and with less systematics. As relaxed clusters are desirable, there exist a plethora of criteria for classifying a galaxy cluster as relaxed. In this work, we examine $9$ commonly used observational and theoretical morphological metrics extracted from $54,000$ Mock-X synthetic X-ray images of galaxy clusters taken from the IllustrisTNG, BAHAMAS and MACSIS simulation suites. We find that the simulated criteria distributions are in reasonable agreement with the observed distributions. Many criteria distributions evolve as a function of redshift, cluster mass, numerical resolution and subgrid physics, limiting the effectiveness of a single relaxation threshold value. All criteria are positively correlated with each other, however, the strength of the correlation is sensitive to redshift, mass and numerical choices. Driven by the intrinsic scatter inherent to all morphological metrics and the arbitrary nature of relaxation threshold values, we find the consistency of relaxed subsets defined by the different metrics to be relatively poor. Therefore, the use of relaxed cluster subsets introduces significant selection effects that are non-trivial to resolve.