Space agencies like NASA define astronaut based on whether someone as completed the training to be an astronaut. NASA also recognizes Astronauts who have flown missions with a gold pin instead of a silver one. This is standard across government space agencies. This is the only reasonable definition of an astronaut. By this definition anyone who must meet any degree of physical, mental, and training standards to fly is an astronaut.
This will always be the case since the acceleration and kinetic energy imparted to the body will always put stresses on the body that could cause medical issues if not death. Space tourist will always need to be reasonably fit in a world where 2/3 of the people are overweight or obese. Space tourist will need to know how to evacuate the craft in an emergency. There will have to be a way out in an emergency once this becomes more of a mass market.
Government astronauts will always be a cut above, but the term astronaut is like the term sailor. One can be a sailor even if they only paddle around on a small lake.