What primarily determines the official distinction between an astronaut and a cosmonaut?

Which nation’s space agency trained and certified the individual

The fundamental criterion differentiating an astronaut from a cosmonaut is not the act of flying in space itself, but rather the governmental or international body responsible for the individual's training and official certification. The title reflects the heritage of the space program that developed the crew member. For example, individuals trained by NASA, ESA, CSA, or JAXA are designated astronauts, whereas those trained by the Russian Federal Space Agency (Roscosmos) or its Soviet predecessor are designated cosmonauts. This historical split originated during the space race rivalry between the United States and the Soviet Union, and the naming convention has persisted as a marker of national lineage, even when crews collaborate.