There aren't any rules in America and there are psychoanalysts who may/may not be psychiatrists. But psychiatrists are all medical doctors, specializing in psychiatry (as opposed to internal medicine, like most primary care physicians (PCP)).

Some psychiatrists talk too, but being doctors, they are more expensive and they mostly prescribe medication and one sees them for 15-20 minutes as one would see one's PCP (I go to my PCP quarterly). It's a medical checkup. Often there are groups, a psychiatrist and several psychologists and social workers and one may see different members of the same group for different "tasks", one for med checks, one for individual psychotherapy, and one for group therapy, perhaps.

Many people have their PCP prescribe their meds, don't even see a psychiatrist. There are others who can prescribe meds, nurse practitioners, for example (but they usually have to be in a group with a doctor to oversee them) and about a zillion different types of people who can do psychotherapy.

The US is a bit like 50 different countries, each state decides how people are licensed to operate in that state and usually one can't practice in another state (though you can drive through one, just can't stay there without giving up your one state's driver's license and getting one from the new state you live in :-) the Federal Government isn't like most other countries' Governments, it just governs things that apply to everyone, like interstate commerce and interstate roads, immigration, censuses/Federal taxes, military/wars, National Parks, etc.

The US doesn't have a national school system, which is why our education/school scores lower than other nations; if you take an "average" of the 50 states that would not be the same as if you took 10 states with "good" education or 10 states with not-so-good and used them. Where they are getting the students taking the tests that decide how the "nation's" education is, makes the difference. The US is so very large, that there are bound to be states/pockets of places that have problems that other states don't have or that value education more and have the resources to back that up, etc. I still vividly remember living in Norfolk, Virginia when I started 7th grade and working on multiplying fractions and then moving after a month to California where I was thrust into the Yale University School Mathematics Study Group and suddenly trying to understand how to multiply base 7 numbers as well as doing non metric geometry; very avant garde for the mid-1960's. My father could not help me with math because he, like all the other adults, had learned plane geometry and had no clue about this "new" math :-)

The main difference between psychiatrists and psychologists is not whether one is better than the other but the thrust of their studies. The psychiatrist, as a medical doctor, is studying how medicine affects the brain and the psychologist is studying psychology, which is about behavior.