There is some dispute over whether schizophrenia is a well-defined construct. Schizophrenia (the called dementia praecox) and bipolar disorder (then call manic-depressive insanity) were originally distinguished from one another based on some aspects of clinical presentation and pharmacological intervention that were apparently relatively arbitrary.
The existences of brief reactive psychoses, psychotic mood disorders, and schizoaffective disorder call the dichotomy between schizophrenia and mood disorders into question.
The forms of schizophrenia would suggest this, which is why I mentioned the broad categories:
-disorganised, catatonic, paranoid, which really are good descriptors, because they don't really match with the words in the case of paranoid.
-disorganized is areal thing these are the people who show major thought disruption and thought disorders
-catatonia is a real thing as well, however not very commonly seen because of medication, it used to be considered to be associated with the degenerative schizophrenias as well
paranoid would be better characterized as 'primarily associated with delusions' and high functioning
the other two (not in the DSM)
-predominant negative symptoms
-affective features
And the other things to consider: there are currently over 300 chemicals implicated in neurotransmission, so there are going to be a huge potential of possible paths to schizophrenia
And as you rightly point out then there are the other forms of psychosis and affective components. I don't think it is a dichotomy in practice, too many clients/patients have mixed symptoms for that. And there are huge battles between psychiatrists over where those lines lie, some use schizophrenia, some use bipolar disorder.
One of my supervisors suggested that ideally the categories would really be spectrum. And you could use a pyramid with four points:
-psychosis
-depression
-mania
-anxiety
And categories people on those four scales, however it leaves out the negative symptoms.
Then there are the weird things like intrusive thoughts in OCD, which look a lot like delusions, but aren't