I just don't like the idea of "If you're bit by a vampire, you become a vampire", because that's not how it worked in Bram Stoker's book. Plus, if that's how it worked, simple math shows you that it would only take a week or so for the entire world's population to become vampires.
In the original, if you were bit by a vampire, nothing special happened to you, you just got drained. You wouldn't even die as Dracula himself would only finish you off if you were a nuisance to him. The only way to become a vampire was to drink the blood of a vampire, and Drac himself recruited vampires by cutting his chest, and holding whomever he wanted to be a vampire up to his chest in such a way that they HAD to drink his blood in order to breath. (Btw, that was the weird sexual symbolism that some people are talking about)
Drac himself became a vampire through some religious mumbo jumbo that wasn't explicitly detailed in the book, but shown in the movie of the book.
Yeah, I haven't actually ever seen any fiction where anybody a vampire drinks from becomes a vampire. The idea is vulnerable to a pretty obvious reductio ad absurdum as you described.
I think my favorite vampires ever were the ones in Vampire$ by John Steakley (which was apparently adapted, not very faithfully, into the movie "John Carpenter's Vampires"). The book itself ain't bad either, the characters are completely over the top - the men are all stone cold badasses with hidden soft spots, the women are incredibly gorgeous and elegant but also extremely tenacious and so on, but they're all developed in ways that are still interesting to read about if not entirely believable.
Anyway, the vampires in that are fucking monsters. They can use mind-powers of a sort to hypnotize people and appear very beautiful, but as soon as the spell is broken people see them as rampaging demonic monsters. They can enslave people by biting them, their transformation stage is apparently accompanied by rotting and being infested with maggots for a time, they can turn people into lesser, zombie-goon type minion vampires, they're superpowerful, and daylight blows them the fuck up, although at the end of the book they use flamethrowers to kill a master vampire at night, which is a big deal.
Anyway, the unadulterated monstrous evil of those vampires is kind of a nice change to the emo bitch vampires of, say, Anne Rice.
An honorable mention, and easily the weirdest goddam vampires ever, would be the ones from the Necroscope books. I didn't think the books themselves were really all that great, but get this: the vampires originate in an alternate dimension, as MUSHROOMS. YES MUSHROOMS. They shoot off these spores that infest people or animals; once infected, a vampire worms grows in the person's body, parasitically. So they sort of become one, but not necessarily completely bonded because the vampire can leave for a new host. They get all kinds of insane powers, like growing and molding their flesh however they want, and creating all kinds of genetic horror monsters as minions this way. The main use, however, seems to be for raping people. When a vampire dies, its body rots into more of the mushrooms and the life cycle repeats.