I think the difference between psychologist and psychiatrist confuses a lot of people. It's not that surprising though, because if you've never had to deal with them before, or it's your first time going down that path, you're not likely to know that.

One thing that it seems must vary from place to place is who is allowed to assess people for certain things and diagnose them. I've seen people over on the ADHD subreddit say they were diagnosed with ADHD by a psychologist but over here in Australia only the psychiatrists can do that, not a psychologist. The medication side of things is the same here as what you've said though.

It seems very dependent on which disorder a person is being assessed for though. My psychiatrist who treats my ADHD says I probably have Autism too but says I need to see a neuropsychologist to be assessed for it, so it looks like who can do what varies a lot. I'm surprised ADHD is a different specialist to Autism here though given they are both neurological disorders and they have a lot of overlap of symptoms. I've even seen discussion of whether ADHD should be considered a spectrum disorder. Idk how this works in other countries but given the similar nature of the two, I find it odd a psychiatrist assesses for ADHD but a neuropsychologist assesses for Autism.