In popular usage, schizophrenic (and the more slangy and now dated schizoid) indicates split between two attitudes.
This drives people with training in psychiatry crazy.
Schizo- does indeed mean split, but it is used here to mean split off from reality.
Someone with a Jekyll-and-Hyde personality is suffering from multiple personality disorder (or, more recently, dissociative identity disorder), not schizophrenia.
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