It appears that operator chairs are a kind of office chair, but I have not been able to find an explanation describing exactly what distinguishes an "operator" chair from any other kind.
Regarding telephone operator chairs, these women were packed very close together in rows. The chairs had to swivel in order for them to get in and out past one another. How do I know? I just asked my mom who is eating breakfast beside me, and was an operator from '48 to '55. They had a back but no armrests.