If you call someone who kills many people in a specific manner a "serial killer", what do you call a murder case like that?

Yes, Serial Murder is an accurate term for referring to several related murder cases at once. It is still multiple cases, though.

Murderpedia calls John Wayne Gacy a serial murderer and a serial killer but says he was arrested for “murder” in 1979. That may be because there was no separate criminal statute for serial murder. If you plan to write on the topic, the Murdepedia might be a useful resource.

Even I think Serial murder is the appropriate term.
“A series of two or more murders, committed as separate events, usually, but not always, by one offender acting alone” or, including the vital characteristics, a minimum of two murders.Most of the killings involve sexual contact with the victim. FBI states that motives for serial murder include “anger, thrill, financial gain, and attention seeking”.

Serial murder has many varying definitions. The FBI now defines it as two or more murders committed by an individual or killing partners with a cooling off period in between murders in which the person resumes his/her normal life. That last part is very important because the cooling off period distinguishes serial murder from spree murder (multiple murders, multiple locations, multiple events, no cooling off period) and mass murder (multiple murders, one place, one time, no cooling off period).

Serial murder does not have to be done “in a specific manner.” Many serial killers have M.O.s that make them more identifiable, but that’s not a requirement to being a serial killer. The smart ones will switch it up to throw off law enforcement.

“Serial killer” is the same as “serial murderer,” so “serial killing” and “serial murder” are both appropriate when referring to these cases. Hopefully that answers your question.

Sorry for being a bit long-winded. Studying homicide is sort of my thing.

@livelaughlove21 it’s new to me, all the information of you. Thanks for enlightening me so much :)

I don’t think there is any difference between serial killer and serial murderer, any more than there is between “killer” and “murderer.” “Serial” itself just means more than one, not that the they are done in a similar fashion. If I shot you and stabbed your brother, I would still be a serial killer/murderer.