The original show in which she was created by Bruce Timm and Paul Dini is Batman: The Animated Series (1992).
There is one episode which delves in details in her background and how she ended up with the Joker, titled Mad Love, but after rewatching it, it doesn't look like the characters are calling out her profession in any way. We only see her arrive at Arkham Asylum to start working on high profile patients, since she believes they are more exciting to work with.
In another earlier episode titled Harlequinade, the following conversation happens (emphasis mine):
Harley: Did you see the way I handled those creeps? Pow! Bam! Batgirl, eat your heart out!
Robin: What was she before she went bonkers?
Batman: A clinical psychiatrist.
Robin: Figures.
So it looks like her background didn't change from psychologist to psychiatrist but the other way around. Or whoever wrote that in Wikipedia chose the wrong vocabulary, if there is no source to support that claim.
That being said, different shows tend to have events happening differently. Usually, it's because of who wrote the story, but I guess it can also be chalked up to "different universes" in-universe.
In the comics, in the canon first appearance, Batman: Harley Quinn (1999), she was recounting her story after being retrieved/saved by Poison Ivy from rubble.
She was described as being an intern at Arkham Asylum, and her doctor's license was revoked when they found out she was helping the Joker escaping from the place. She also briefly said she had "a couple years of med scool" when asked for help in saving injured civilians.
Then later on, Batman called her a demented therapist.
I don't know much about the difference between a psychologist and a psychiatrist, but a quick search seems to bring up that licensed therapists need a Master's Degree, whilst psychologists would have a Ph.D instead, which seems to fit with your description.
So far, it looks like in both animated series and comics adaptation, she was never described as being a psychologist in the original origin stories.