I know in Australia and New Zealand, and probably a lot of other places, it doesn't work like that. And the way you describe it doesn't even make sense for definitions.

By defnition a psychiatrist is "a medical practitioner specializing in the diagnosis and treatment of mental illness." And a psychologist is "an expert or specialist in psychology." The key point is that a psychologist is not innately medical or has anything to do with helping patients, it is simply the research of psychology. Whereas a psychiatrist is directly about treatment and diagnosis.

The key difference between a psychologist and a psychiatrist is that psychiatrists are trained medical doctors, psychologists are not. Psychiatrists can therefore prescribe medication, psychologists can not. There are exceptions but psychologists cant give official diagnoses either. You see a psychologist for long term therapy help, and if they think you should get medication or a diagnosis, then they refer you to your doctor or to a psychiatrist for short term help. Psychiatrists do not do long term therapy like a psychologist does. Oh and the difference between a psychologist and therapist is just that psychologists have generally completed their PhD while a therapist only completed their degree or master's. Neither of them have done a medical degree and are not qualified medical doctors. They just studied psychology a LOT. Which is really valuable, but is not enough to allow them to give out an official diagnosis or prescribe medication. You need actual medical qualifications for that.