Charlie, that more or less covers it, I think.
Transsexuals go to doctors for help because we KNOW something is wrong with us, because we accept the reality of what our biology says, and want to modify it as best as possible to match what we somehow feel instinctively inside it ought to have been. So in any language that needs medical assessment and help to resolve in the way best determined by science after a thorough investigation of why you have what is self evidently something that has gone awry .
The wider trans umbrella sees it more as a form of self expression, lifestyle choice, or just wanting to live as who they feel they need to be to feel better than now and without in many cases suffering any significant dysphoria. Indeed, many will be older and so will presumably have coped with their condition for years which most transsexuals struggle to do past childhood.
So extreme medical intervention is not sought if the dysphoria is not there.
The GRA today only allows legal recognition for someone with dysphoria who is willing to go through the medical and psychiatric gatekeeping process and fulfil the other conditions such as a two year hold to establish that transition has been successful.
The push for self ID to lead to legal recognition comes from those who do not believe they have any medical condition. so do not need to see a doctor, or get medical help, as they believe it is a free choice to express who they feel they are.
Theresa May has backed this by saying being trans should be demedicalised.
That about covers it I think. Though someone will hopefully correct me if I am misunderstanding anything as I only really 'get' the transsexual part from direct experience.