You are partially correct actually. The difference is that the original, actual zombies were controlled by a human "sorcerer" or a bokor and not a demon (or a Fallen angel). They were usually re-animated corpses but at times living people were turned into zombies as well, which pretty much made them mindless puppets and they were almost never dangerous (unless they consumed salt). What really happened though is that they were drugged by certain toxins. This is more or less today's medical lobotomy.
It does sound similar to the [REC] universe and Tristana controlling all of those she posses (infects). However though, Tristana is possessed by an unspecified demon which means that the demon itself controls all of the "zombies", making them demonically possessed and not zombies by either Hollywood or African Voodoo concepts.
Also, maybe you didn't really go in-depth of [REC]'s story but another interesting thing is that originally Tristana was a Christian and even lived in a convent. And due to Christian beliefs, true Christians cannot be possessed by demons because "the Holy spirit already lives there". However, some can become possessed if they suffer from certain emotional trauma, have participated in unholy rituals or were sexually assaulted. The third one in Tristana's case occurred - she was raped by priests. Which goes further to prove that this is a classic possession tale described many times in the Bible and has nothing at all to do with zombies of any sort and lumping them together may be considered offensive by some.
Possession and zombification are two different things, albeit originally they were somewhat similar. Possession and exorcisms were introduced LONG before zombies were ever thought of and were even referenced in the New Testament. The only common things the possessed from [REC] have to do with the Hollywood zombies is their appearance and the way of the possession spreading (virally).
And besides, following your logic we can also assume Regan from The Exorcist was a zombie.