Yeah, I'm curious, why do we even do this?
The psychopharmacology era came about during the psychoanalytic heyday of the mid-20th century. As such, in locales like NYC and Boston where psychiatry was primarily psychoanalytic, "psychopharmacologist" was used to distinguish psychiatrists who specialized in drug treatment from those who were primarily therapy based. While American psychiatry was readily accepting of antipsychotics for schizophrenia, the use of minor tranquilizers and antidepressants was much more controversial. Most of the psychoanalysts were very much against drugs for neurotic disorders. I remember my former chair recalling being criticized as a resident by his supervisor for prescribing Elavil to a depressed patient being told "what your patient needed was an interpretation, not a medication." Split treatment, with patients having a psychoanalytic psychiatrist and a psychopharmacologist was common in certain locales.
For many biological psychiatrists of the era, psychopharmacology (which was a legitimate science and active area of basic and clinical research) provided a legitimacy to the field and its scientific underpinnings. There was real optimism that understanding the mechanisms of the drugs would provide insights into the pathogenesis of mental disorder (and in some cases, it did). Psychopharmacology also brought the randomized control trial to psychiatry. It was the psychopharmacologists who made CBT (or cognitive therapy as it was then) successful by trying to package it and test it like a drug.
Societies like the ACNP and CINP were very important in promoting the field of psychopharmacology. In the early days it wasn't so corrupted by pharma and even the drug companies were genuinely interested in knowing which compounds might have therapeutic use. The field as an academic discipline was very multidisciplinary including psychiatrists, psychologists, biochemists etc.
There was also a significantly different body of knowledge associated with prescribing medications compared to dynamic psychiatry. As such, psychiatrists committed to learning the intricacies of drugs' putative mechanisms, interactions, cautions and contraindications, pharmacokinetics and dynamics, and toxicity. Psychiatrists were far more likely to know their drugs (it wasn't a large number and many were "me toos") inside out compared the average internist or most other medical specialists. That is probably still the case today. Perhaps with the lack of certainty of so much in psychiatry, the known facts about the drugs we used provided some comfort to clinicians and served as foil for those who thought of psychiatrists as only wooly minded navel gazers whose practice was no more scientific than astrology or tea leaf reading.
Personally, I think the terms psychopharmacology/psychopharmacologist are infinitely preferable to terms like "medication management" and "prescriber." I suspect the reason it can sometimes feel a bit cringe for clinical psychiatrists to call themselves psychopharmacoogists is because it was in part a form of overcompensation.