Step 1: Trace the term to its inventor. Psychiatry as a named medical discipline first appears in the early nineteenth century.
Step 2: The credit belongs to $Johann\ Christian\ Reil$, who introduced the word in 1808 to describe the medicine of the mind.
Step 3: Distinguish the others: Bleuler is tied to schizophrenia and the four A's, while Pinel pioneered humane moral treatment of asylum patients. Neither coined psychiatry.
\[\boxed{Johann\ Reil}\]