Interesting thread. I used to work with a lot of psychiatrists and psychologists. Many were lovely people, some weren't, much like any other walk of life. Some were so lacking in empathy that it was a mystery to me how they'd ended up in a caring profession. I do think there's something in the idea that for some psychology/psychiatry is a draw because they haven't a clue how normal people think/feel. Also, some of my ex-colleagues had very difficult family backgrounds, family members with addiction problems etc, and their career choices had been influenced by a desire to do what they could to help others with similar problems.
As far as I understand it, a counsellor is someone you go to to talk through personal difficulties, which won't necessarily be classed as mental health problems - e.g. relationship problems, coping with bereavement, family stress - although obviously if bad enough any kind of problem can lead to a mental health problem like depression or anxiety. They should be non-judgemental and don't tell you what to do, but they guide you to think through your problems and work out coping methods. I've not seen one, this is just based on having read about it - is this accurate?
I think a lot of people get confused between psychotherapist and psychoanalyst (I certainly used to). Psychoanalysts pop up in old films and books, and are psychiatrists or psychologists who've trained in Freudian or Jungian analysis, which would involve having extensive therapy themselves, and their patients lie on a couch talking about themselves (in films, anyway). That kind of analysis takes years and years so has tended to be the preserve of the wealthy but worried well. Uncommon now? I don't think it has a very good success rate in helping feel better about themselves, but am prepared to be corrected there!
Psychotherapists might be psychiatrists but are more likely people with psychology degrees who've had some postgraduate training in administering a particular therapy, like CBT, or a range of therapies. They might be clinical or counselling or health psychologists. If I understand correctly, the therapies involved take a lot less time to administer than traditional analysis, so are far cheaper. Research seems to indicate that CBT is quite successful at helping people with various common mental health problems, when used in conjunction with a short course of medication.