The word zombie is not in the Romero trilogy that started this genre, but I know at least by the late 80s Dawn Of The Dead was called the first zombie apocalypse movie.
I suspect like a lot of supposed tropes of these movies it came from Return Of The Living Dead, also the source of the brain eating trope which was never present anywhere else.
Prior to Romero, “Zombie” was a different sort of monster – an animated corpse brought back to life via voodoo. They shambled along, but were under the control of a voodoo master. They did not eat brains.
While Romero’s living dead were not called zombies, the similarity to the older version probably caused people to use the shorter term. In addition, “zombie” had already meant someone who did things mindlessly.
It appears that the eating of brains does come from Return of the Living Dead
Although Romero didn’t use the word “zombie” in his “living dead” films (he called them “ghouls” or “living dead”), the international title of his 1978 sequel, Dawn of the Dead, was Zombi. It was quickly followed by a non-Romero “sequel”, Zombi 2. So that’s where the word first got applied to the people-eatring variety of living dead.
As noted already, “Zombie” originally applied to non-threatening reanimated walking corpses of the Haitian variety. Actually, although they didn’t try to eat you, they were depicted in some early films as being perfectly capable of killing people. They just didn’t get anything out of it.
George Romero admitted that he was inspired by Richard Matheson’s I am Legend, which you’d think would make his “ziombies” lineal descendants of vampires . I’ve long argued, however, that Romero’s film actually has more in common with the 1959 movie Invisible Invaders, which features corpses re-animated by invading aliens (a la Plan Nine from Outer Space, but with MUCH better script and production values). The images of the “zombies” (not so called in this flick) resemble the “ghouls” in Night of the Living Dead – dressed as they were (men in business suits), with black-rimmed eyes and slow, stiff gaits. And their extraterrestrial animation is similar to what is hinted at in Romero’s first film – an animating influence being broadcast by a satellite in space.
Just to add: Although Dawn of the Dead wasn’t called “Zombi” in the US, the “sequel” Zombi 2 certainly was – In fact, since there was no “Zombie 1” in the US, they just called it Zombie. I remember seeing posters for it and reviews in magazines like Cinefantastique in 1979.