Perhaps. In Bram Stoker's book "Dracula", which is where the legend was popularized, Lucy becomes a vampire after being bitten and drained over several weeks until she finally "dies", is buried, and then comes back from the grave as a vampire. It's not said whether she ever drinks of Dracula's blood. On the other hand Mina does drink from a self-inflicted wound on Dracula's chest on one occasion, though she doesn't become a vampire and that might be symbolic of drinking from the breast of one's mother, showing Mina's loyalty to Dracula. I'm sure I'm confusing the stupid Keanu Reeves movie with the book though which I haven't read in a while and have seen the movie a couple times since.
But certainly Anne Rice's vampires require draining and drinking of the vampire's blood before they become vampires themselves. Just biting is not sufficient as they tend to blood-lustfully kill their victims rather than say, drink a pint or two and let them live. Most vampire movies follow these rules now-a-days, which sort of extrapolate what appeared in "Dracula" and was spelled out by Anne Rice. "True Blood" for example follows the same rules with a little extra ritual tacked on.
If you want to know the simple origins of vampire/zombie legends, which were one and the same monster originally, it's a fact that an untreated corpse will sit up after a while, and of course everyone knows the fingernails and hair seem to grow as the skin pulls back in death, and out of fear people decided the only explanation was that the bodies weren't entirely dead, and so rituals were invented to keep the dead resting, such as garlic in the mouth or coins on the eyes, prayers, and whatnot.