The first "zombie apocalypse" movies (as opposed to a few actual "zombies as the slaves of a voodoo witch doctor movies) always hinted that it was a supernatural phenomenon, complete with some elements of a Christian rapture where the dead walk the Earth.
In the original Night of the Living Dead, they're in a cemetery when the zombie attacks, having presumably just dug itself out of a grave or perhaps crawled out of an unfilled grave. Very soon though, we start seeing science fiction elements to these movies, even Romero adopts this tact. I haven't been able to nail it down to the first that does this, but certainly by the 1970s this is the case. By the 1980s, many are considering the cause to be any or many of the following:
- Biological weapons that have malfunctioned or been tampered with
- Chemical weapons that have malfunctioned or been tampered with
- Herbicides that have been tampered with
- Exotic radiation
- Strange pathogens/substances that have arrived from spaceships or meteorites
Often, these explanations are not explored in depth, and come off as highly implausible. Worse, they tend to depict phenomenon that could never be plausible in the real world. Already on The Walking Dead we've seen putrefaction and damage that would make it impossible for the cadaver to be mobile at anything more than the pace of a slime mold. It takes constant and high levels of energy to move a human body around, and no blood is pumping oxygen and nutrients (not to mention the state of decay of muscles).
But a supernatural explanation would be unpopular outside of tiny niches. Supernatural explanations always seem to retreat back to Abrahamic mythology, and I suspect rather strongly that no one wants to go there. (The television show Supernatural does just this, and while people enjoy it, it's somewhat niche.)
The new variation where you aren't killed by the pathogen, but become a zombie after something else kills you... I'm fairly certain this starts with The Walking Dead, and then only in later seasons. In season 1, the black man tells him that when bitten you get a fever and die in a matter of days.
The recent remake of Dawn of the Dead however has this happening, and much more quickly. The sickness doesn't seem to last longer than 48 hours, and is highly transmissible (a scratch is enough).
I don't watch any of the really schlocky movies, so I might have missed a few. But I'm going to say this is The Walking Dead, and it only (so far).