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.