Around here when you commit a crime you'll be evaluated whether you are accountable, diminished accountable, or not accountable.
Accountable: you'll get a sentence and you'll serve that.
Diminished accountable: you'll get a shorter sentence and then you'll go to the clinic for people who are not accountable and are a danger to others, and you'll be released when you're not a danger to others anymore. (Shorter sentence: say you're supposed to get 10 years, but the judge says you're only 50% accountable, then you'll have to serve 5 years. Then to the clinic.)
Not accountable: to the clinic until you're not a danger anymore. ("Insanity defense")

The clinic doesn't have a predetermined stay lenght (although after 6 years, you're moved to the longstay) - you'll just stay until you're safe again. For some, that means they'll never leave. I think it's evaluated yearly for the first few years and then every two years.
This also means some people refuse to cooperate in any way with their evaluation, afraid they'll be sentenced to indeterminate clinic time. For example - if you do something that might get you 4 years and you'll be declared not accountable (I think you US lot call it insane) you can be moved to the clinic instead, but you might have to stay there 10 years.

So in my country people 'psychotic murderers' aren't sent to a regular mental hospital, but to a clinic that houses only other mentally ill criminals.
If you commit a crime that would get you less than 4 years, and you're mentally ill, you'll generally not be sent to the clinic unless it's a specific crime (I think stalking is one of them, and a few others). Maybe then you'd be sent to a regular locked ward - not sure, I'm no lawyer.