TL; DR: No. Bram Stoker introduced the idea of a vampire's powerlessness before a crucifix in Dracula. For centuries (perhaps millennia) before that publication in 1897, however, people killed vampires with stakes to the heart.
Scholars believe that humans recognized the existence of vampires even in prehistory. Archaeologists have pulled prehistoric mummies out of peat bogs that the archaeologists hypothesize were executed for being vampires.
When Danish historian Saxo Grammaticus published Gesta Danorum (between 1180 and 1220 CE), he noted that Finnish people attacked and slew Odin, after which Odin caused "abominations" and "pestilence" to spread until the locals removed his body from his burial mound, beheaded it, and drove and sharpened stake through his heart. Grammaticus tells at least one other story of a person rising from the dead in his opus. Again, locals solved their problem by staking the demon through the heart.
Romanian folklore
Ion Creangă was a defrocked Romanian Orthodox priest who collected Romanian folklore and wrote short stories between 1857 and 1889. Here are some quotes from his collections of folklore:
In "Amărăşti, a rural village in central Romania, an old woman died. After several months the children of her eldest son begin to die, and then those of her youngest son. The sons go to their mother’s grave, dig up her body, cut her in half and re-bury her. Still the deaths continue. Once more they dig up her body and are horrified to find it whole again, so they take the body into the forest, where they cut out her heart and burn it, mix the ashes with water and give the liquid to their remaining children to drink. After this, the deaths cease."
Here is another story from Ion Creangă:
"A final example, also from Siret, [Romania,] details how some returning soldiers are offered a lift by an old man on his cart. As they travel through Transylvania, where the man was attempting to buy some hay, they stop at a house where a woman invites them in and gives them a meal of porridge and milk, before leaving them to eat in peace. After the meal, the soldiers look for the woman to thank her but cannot find her anywhere. On climbing into the attic, they are horrified to discover the bodies of seven people, one of which is the woman who made them the food. The soldiers flee in fright, but as they do so they see seven lights descending on the house – the souls of the vampire bodies! The story narrates how, if the soldiers had turned the bodies over so they were facing downwards, the souls would not have been able to re-renter the bodies."
This is a quote out of From Demons to Dracula: Creation of the Modern Vampire Myth that summarizes the various ways that Eastern European peasants killed vampires:
"The most common worldwide idea of how to destroy a vampire is to put a stake through its heart. In Bulgaria, though, it should be a redhot iron that is used rather than a stake... In Romania, however, the traditional method is to exhume the body on a Saturday, then drive a stake through the navel or cut out the heart. The heart should then be burned, with the ashes collected, or boiled and then cut into pieces. These are sometimes thrown into a river, but are usually mixed with water to give to sick people or the vampire's relatives. Other methods include laying the body face down in the grave so it cannot rise, placing items in the mouth such as pebbles, garlic, or incense for the vampire to [gnaw], or filling the coffin with millet grains or seeds for the vampire to count, all of which delay the vampire rather than destroy it. Garlic ... has powers to ward off evil spirits, vampires and wolves, and on particular days of the year when it is thought evil spirits are abroad, garlic is rubbed on windows in the pattern of a cross, placed above the door or rubbed on all means of entry to a building, and sometimes even rubbed on farm animals to protect them against vampires."
John William Polidori's The Vampyre
In 1816, physician John William Polidori and his employer, Lord Byron, went away for a stormy weekend retreat with a veritable "who's who" of Romantic literary stars: Mary Wollstonecraft Godwin (better known by her married name, Mary Shelley); her fiance, Percy Bysse Shelley; her stepsister Claire Clairmont; and Lord Byron. These five held a story-writing contest at their villa. On that same day, Mary Shelley wrote a story that would become Frankenstein, and John William Polidori wrote The Vampyre, the first English-language story in the vampire genre.
In The Vampyre, a mysterious, Byronic, glamorous, Greek aristocrat named Lord Ruthven stymies the protagonist, a wealthy young English gentleman named Aubrey. Aubrey befriends the mysterious aristocrat and travels with him to Rome and Greece. Along the way, Aubrey learns tales of vampires. At the same time, beautiful, virginal young women mysteriously die all along their route. Later, bandits attack the pair and Ruthven dies. Aubrey heads back to London, where he re-meets the very much alive and well Lord Ruthven! Not only is Ruthven alive, but he is actively seducing and now engaged to marry Aubrey's sister. (Aubrey's sister never gets a name; she is only important in how she is connected to the men in her life.)
I will not spoil the end of the story, but if you like the gothic, spooky beauty of Frankenstein and stories of Byronic brooders, I highly recommend The Vampyre. You can read it for free on Project Gutenberg! It is a short story that you can read in less than an hour.
Notably for the purposes of this question, the first English-language vampire story never mentions crosses, communion wafers, or even garlic or stakes to the heart.
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