Everything I need to know about life,
I got by killing smart people and eating their brains.
-- anonymous
Though they both exist in the popular consciousness, the distinction between zombies as brain eating or flesh eating creatures stems all the way back to the making of Night of the Living Dead.
The film was written by George Romero, who became an icon in the history of horror cinema, and John A. Russo, who's slightly less well known. After the film was finished, the two men disagreed on which direction to take the intellectual property they had created and went their separate ways.
Russo wrote a novel called Return of the Living Dead, later made into a film by Dan O'Bannon, where for the first time, zombies craved exclusively brains. Romero went on to make a whole series of movies where the creatures ate any part of someone they could grab, all with titles ending in "of the Dead," not "of the Living Dead," and with that single word of divergence, there was no longer a single authoritative source to determine what modern zombies were and were not, and the chaos could only continue to grow.