Consider therapists who have you lie on a couch vs those who have you sit in the same style chair, as an equal.
Some therapists position themselves as an authority figure for their patients. If they were to open up about themselves, the patient might think they're incapable after seeing the therapist has doubts in their own life.
This has advantages and drawbacks. It may be more efficient in earning trust, thus helping some patients heal faster. OTOH, some patients may never really accept this kind of therapist.
In my experience, psychiatrists more often position themselves as authorities, and among psychologists/therapists/counselors there is a mix.
It's really hard IMO to get communicable science out of talk therapy because there are so many variables and it is extremely subjective and personal. The whole point is to help a patient navigate their own issues.
I draw the above opinion on therapy from reading a few books about the history of psychoanalysis and the various branches it has taken.