You claim that "mental illness in general is more about mood".
No, technically speaking, mental illness is an umbrella term, which also encompasses mood disorders. The very quote you used to base that conclusion on says
Mental illness, also called mental health disorders, refers to a wide range of mental health conditions — disorders that affect your mood, thinking and behavior.
The fact that they listed mood first doesn't really mean any prioritization of mood over the other factors, in the scientific literature at least.
If you want to refer to the use of this "mental illness" term by the general public, it's probably more likely to be understood as referring to psychotic disorders.
Results indicate that the majority of the public identifies schizophrenia (88%) and major depression (69%) as mental illnesses
Not including personality disorders in the list of examples is perhaps a confusing oversight on that Mayo page for "mental ilness". Also, note that the more official DSM terminology speaks of mental disorders not mental illnesses. And personality disorders are certainly included e.g. MedlinePlus says:
What are some types of mental disorders?
There are many different types of mental disorders. Some common ones include
- Anxiety disorders, including panic disorder, obsessive-compulsive disorder, post-traumatic stress disorder, and phobias
- Depression, bipolar disorder, and other mood disorders
- Eating disorders
- Personality disorders
- Psychotic disorders, including schizophrenia
As discussed in Poidah's answer, this strict/conventional categorization is somewhat debatable.
As for why this "personality" terminology... it is due to Kraepelin. Even though his list of personality disorders bares little resemblance to the modern (DSM) one, his idea that there's a spectrum of personality traits that spans the healthy and the abnormal has persisted... Along the way came some psychoanalytic ideas, which shaped some of the categories still used today (despite the subsequent [near-]demise of psychoanalysis from the mainstream psychiatry). In (a lot of) detail:
Emil Kraepelin (1856-1926) introduced personality types into modern psychiatric classification, under the term “psychopathic personalities.” [...] Kraepelin stressed the existence of a broad overlap between overt pathological conditions and personal features that are encountered in normal people. He noted that the limit between pathological and normal is gradual and arbitrary. In entering the field of personality, psychiatry was taking an interest in conditions that were not previously considered to be liable to psychiatric interpretation. In the 7th edition of his textbook, Kraepelin assumed that psychopathic personalities were the consequence of a faulty constitution, which had previously been approached under the ill-defined concept of degeneracy. Psychopathic personalities result from a psychological inborn “defect,” which explains why the symptoms of psychopathic personalities have always been present in the individual and persist with little modification during his or her whole life. Their pathological nature is not deduced from the fact the symptoms appear in the patient after a period of normal functioning, but rather from the fact that symptoms deviate from the range of normalcy. Patients with psychopathic personalities often have good cognitive capabilities, but their affects and emotions are problematic. In the 7th edition of Kreapelin's textbook, the list of pathological personalities comprised only four types: (i) the born criminal (der Geborene Verbrecher), modeled on earlier description by Cesare Lombroso (I'uomo delinquente) and James C. Prichard (moral insanity); (ii) the irresolute or weak-willed (die Haltlosen), who are unable of applying themselves to sustained and long-term work; (iii) the pathological liars and swindlers (die krankhaften Ttigner und Schwindler) whose disorder is due to hyperreactive imagination, unfaithful memory, an unstability of emotions and willpower; and (iv) the pseudoquerulants (die Pseudoquerulanten) who correspond to today's paranoid personality. The prefix “Pseudo” was meant to differentiate this personality from the delusional disorder of the same name. In the 8th edition (1915), the list was expanded to seven types: (i) the excitable (die Erregbaren), possibly sharing some characteristics with today's borderline personality disorder; (ii) the irresolute; (iii) persons following their instincts (Triebmenschen) such as periodic drinkers and pleasurelovers; (iv) eccentrics (Verschrobene); (v) pathological liars and swindlers; (vi) enemies of society (Gesellschaftsfeinde); and (vii) the quarrelsome (die Streitsüchtige). Kraepelin studied patients whose symptoms had consequences on social adaptation, and for whom a psychiatric opinion might be sought after some problem with the law. Most of Kraepelin's personality types do not correspond to DSM-IV-TR categories.
[...] DSM-II (1968) was influenced by psychoanalysis; in DSM-II, some personality disorders had to be differentiated from the neuroses of the same name (eg, hysterical, obsessive-compulsive, and (neurasthenic personalities and neuroses). In DSM-III (1980), and the subsequent DSM-III-R (1987) and DSM-TV (1994), personality disorders were described as discrete types, grouped into three clusters, placed on a separate axis (axis II). This categorical approach was in line with the medical model advanced by Emil Kraepelin. Borderline and narcissistic personality disorders, which entered DSM-III, were adapted from psychoanalytical concepts. The preparation of DSM-5 questioned the merits of combining typological and dimensional models of personality, reopening a century-old debate.
As for another claim of yours that CBT is (as) effective for personality disorders as for mood disorders... I doubt it given that personality disorders in general have a reputation for being hard[er] to treat (by any means). At least the evidence base is rather lacking.