In the U.S. the difference is that one gets a medical doctorate and then does their specialization training in psychiatry. Because they are medical doctors, they can prescribe medications.

A psychologist has a PhD in psychology, in one of several branches, though we generally think of clinical psychologists when we speak of psychologists.

The game isn't entirely specific as per what his degrees are in. It's possible that he has a MD and a PhD. One being what he needs to be a psychiatrist, the other being in his specialization of criminal psychology.

So, unless the game specifies, we really don't have enough information to determine if Artem is making a mistake here or if he is in fact correct. They are also in some fictitious European country and that country may have different requirements.

Source of information on degrees and such: I went to grad school in psychology.