I have no professional insight, but categorizing the different types of people who claim to be trans is a major interest of mine, not least as - IMHO - once we categorize them successfully it is 100% clear that none of them should ever be affirmed. (TLDR - it is all mental illness or paraphilia or dishonest predators, and why should we punish the mentally ill by affirming, or elevate paraphiliacs and predators by affirming. I accept some people claiming to be trans are probably more than one of these things, and some are mentally ill, paraphilic predators.)
From a quick google...
Body Dysmorphia - a mental condition characterized by extreme preoccupation with self-perceived defects, flaws or imperfections in the appearance of the body; body dysmorphic disorder.
Gender Dysphoria - is a state of profound unease, unhappiness, or dissatisfaction, the opposite of euphoria. Term first used in 1973, term first used diagnostically in DSM-5 in 2013.
Body Dysmorphia does seem like a real thing... but then again how many women have it compared to men, and is it not possible that they are all apotemnophiliacs (the paraphilia where a person has a sexual interest in being an amputee)?
Either way, it is absolutely clear to me that body dysmorphic people should not have legs or arms amputated under any circumstances. If other mental health interventions fail then secure accommodation where they can be kept safe is the only compassionate option.
From what I can make out gender dysphoria is just the new name for gender identity disorder, which could equally be called "sex-specific body dysmorphia" (the defect being the wrong-sexed body) or "sex-stereotype dysphoria" or both dependent on whether the person is rejecting the body or the social role. I can imagine pubescent girls might often have both, with the sexual attention and harassment being as a result of girl's bodies and the female social role / place in society.
Sex-specific body dysmorphia is clearly a subset of body dysmorphia and requires MH support, never medical interventions.
"Sex-stereotype dysphoria" should surely be treated in no small part by telling girls that there is nothing wrong with playing football or climbing trees or getting muddy - that they do not live in 1950s USA suburbia and that they only have to look back to some of the icons of the 1970s and 1980s to realize that gender non-conformity is fine.
The whole thing is a massive bait and switch.
According to TRAs (if you actually go behind what they say) "this person does not have a form of body dysmorphia, what they have is dysphoria around regressive sex-based stereotypes". Instead of telling them they can be whoever they wish (ie as gender conforming or non-conforming as they wish) within their unchanging sex we will tell them that medical (hormonal and surgical) treatments, which would never be given for body dysmorphia, should be given. And they should be given because when performing wrong-sex sex-based stereotypes it is easier for the trans person if people pretend they are the opposite sex and if the trans persons body is medicalized to become a facsimile of the opposite sex.
I also have a lot of sympathy for the idea (Dr P?) that for some or many it is Trans Ideation Disorder - where a variety of MH issues such as depression, autism, sexual trauma come together to create someone whose answer is "I'm trans". I suppose in theory TID is the main thing with young trans people, but that sex-specific body dysmorphia and sex-stereotype dysphoria are also useful concepts that can feed into helping to fully understand a young person with TID.
The above is my best, amateur, analysis if we come at it from the position of "let's really try to look in detail and try to categorize all the different sub-sets". But from speaking to someone I know who is fairly senior in mental health in the NHS I am also increasingly coming to share his view. His view - if I understand it correctly - is that a MH diagnosis is not necessarily that helpful. It is not really that useful to label someone as autistic or gender dysphoric. What is useful is to understand precisely what they think and what their issues are, and to try to help them overcome their specific issues. And often the fundamental issue is the reality of their life. For example one can find someone deeply mentally ill, and have them in supported accommodation as part of their treatment. Their MH improves massively, and they are force to leave the supported accommodation, leaving them to move into a horrible house-share with a violent alcoholic at which point their MH spirals out of control again.