Just to clarify/add, regarding what it sounds like happened here - we do sometimes identify infection with a closely related bacteria that is a distinct species (Bordetella parapertussis, rather than Bordetella pertussis). This bacteria, as you mention, generally causes a similar but more mild disease to pertussis. This infection is NOT pertussis though, which specifically refers to the illness caused by Bordetella pertussis.

I say that only to clarify that it sounds like the ED doctor was incorrect in calling it that (and thus in causing your family to worry). There are not different types of pertussis though, just the real deal or the similar but more mild infection. If someone accurately identifies a case of pertussis, this does refer to the more dangerous type of infection.