Context. The generally accepted representation of κ-distributions in space plasma physics allows for two different alternatives, namely assuming either the temperature or the thermal velocity to be κ-independent. Aims. The present paper aims to clarify the issue concerning which of the two possible choices and the related physical interpretation is correct. Methods. A quantitative comparison of the consequences of the use of both distributions for specific physical systems leads to their correct interpretation. Results. It is found that both alternatives can be realized, but they are valid for principally different physical systems. Conclusions. The investigation demonstrates that, before employing one of the two alternatives, one should be conscious about the nature of the physical system one intends to describe, otherwise one would possibly obtain unphysical results. © ESO, 2016.