A psychiatrist was an individual that was proficient in the field of psychiatry. Unlike a psychologist, a psychiatrist was a medical doctor who specialized in treating psychiatric disorders and mental illnesses.
Vulcan Dr. Fer'at was a psychiatric analyst in 2153, who came on board the Enterprise NX-01 to see Jonathan Archer, under the false pretense of being a pathologist checking Archer for pyretic radiation exposure, when in reality he was checking Archer's mental state. (ENT: "The Expanse")
According to Starfleet's personnel file on Elizabeth Dehner, her interest in her own ESP abilities were, in part, responsible for her pursuit and vocational training as a psychiatrist. She also possessed a PhD. (TOS: "Where No Man Has Gone Before")
Doctor Leonard McCoy told Edith Keeler during his visit to 1930 that he was "a surgeon, not a psychiatrist" when she asked if he wished to see James T. Kirk, after he told Keeler he was from the future as well. (TOS: "The City on the Edge of Forever") McCoy, nevertheless, took a stab at psychiatry in 2267, when he tried to convince Eleen to keep her unborn child. When she ambushed him and escaped from his care he decided that "I guess I'll forget psychiatry, stick with surgery. I really thought she'd learned to want it." (TOS: "Friday's Child")
While discussing how Hikaru Sulu had time to start a family, namely having his daughter Demora, Montgomery Scott told Kirk that "if something's important, you make the time." Scott followed by asking Kirk if he was so restless during the dedication of the USS Enterprise-B because he was "finding retirement a little lonely?" Disheveled by Scotty's comment, Kirk responded stating, "with that kind of tact, I'm glad you're an engineer, you'd make a lousy psychiatrist." (Star Trek Generations)