It's not hilarious the mix up - it's a sad state of affairs. Here's some clarification though.
Psychiatrists are medical doctors trained through the NHS. They studied medicine and instead of specialising in oncology, or internal medicine or ob/gyn they specialised in psychiatry. You cannot call yourself a psychiatrist without medical specialisation training. There is no requirement to have personal therapy as part of training.
Psychology. A clinical psychologist (or counselling psychologist) is post graduate training to doctorate level. They have the Dr title which comes from clinical and often research training, but they are not medical doctors. There are other titles like Assistant Psychologist. They are also qualified through NHS training and/or in NHS training. You cannot call yourself these titles without certain qualifications. Psychologists work differently depending on their level and setting, but in general they do talking therapies. There is no requirement to undertake any personal therapy as part of qualification.
Psychotherapy: this is only talking therapies. The title is not protected in the U.K. though so anyone can call themselves a psychotherapist. Usually they have done a private post graduate training (masters) in one of more modality - the training isn't through the NHS. These trainings usually require them to undergo personal therapy themselves. Can be a member of the BACP if accredited training has been completed.
Counsellors People calling themselves this (in the U.K.) may be psychotherapists and have done similar masters level training. Again it's not a protected title though so they could also have done no training, or a course over a few weekends. Good courses will require personal therapy. No NHS training to be a counsellor. Can be a member of the BACP if accredited training has been completed. NB a counselling psychologist is highly trained and is the same level as a clinical psychologist. Not to be confused with someone who calls themselves a counsellor.
The difficult thing is that the type of qualification does not mean someone is better at actually helping people. There are some amazing counsellors and awful psychiatrists or clinical psychologists. Furthermore, certain clinics have frameworks of best practices, that themselves may be quite restrictive (ie only offering CBT), so the professional has to work within that framework. They may be different in their private clinic where they can offer other modalities. However, especially for the last two categories, membership of a professional organisation like BACP means they've done more extensive training than the short courses.
There's more that can be said, but this is just a (sort of) short overview.
Oh and all this only applies to the U.K. online you can find more info but if you're reading an American page, the job titles can be applied differently. A psychiatrist is always a medical doctor though.