I think there's been a shift in media here over the years. Older vampires, the formalwear "I vant to suck your blaaaad" types, often have very long teeth and leave only a couple of punctures after biting someone. It's pretty reasonable to assume these are actually drawing the blood through their teeth like little syringes.

Then as time goes on, vampires have gotten grittier and gorier. We've gone from beautiful damsels with tiny pinpricks on their necks, to staight up cannibalism. Vampires now just fuckin eat people sometimes. Somewhere in between, we did get a phase of vampires biting open the neck with a big gush of blood, and the now classic image of a vampire with blood all over their lower face. At this point it's clear they're just drinking it normally.