Something which I think may be true is that 'Service User' might be more appropriate in a modern context because of the sheer breadth of what is offered in a mental health context.
If I take the hospital in my area which is the main centre for mental health as an example, it's clear that the mental health provision is a kind of massive umbrella - in the middle of the umbrella, you do have the sphere in which people are either on the wards or "attending clinical appointments where they are being prescribed medications, having their mental capacity assessed/monitored and offered psychological therapies", as you describe it, but the edges of the umbrella are about something different, they encompass things like, say, Saturday Brunch For Men; these services are in many ways prophylactic and are intended to be supportive of good mental health in the hope that this provision will obviate the need for people to move closer to the centre of the umbrella. It's when you consider these ancillary, mental health supporting organisations that you can see why 'Service User' might be more appropriate than 'patient'.