It's the lower that is registered as an SBR (and to build an SBR you'd use a Form 1; a Form 4 is for the transfer of an existing SBR). The barrel is just a part, whereas the lower is legally the firearm. And a short barrel isn't necessarily for an SBR, it's completely legal to put it on an AR pistol without any federal registration.
But here's where it can get tricky: If you have the parts required to make an SBR and there's no other explanation for you having them, then you could get in trouble for "constructive intent". Basically, they'd say you intended to make an unregistered SBR. So if you own a short barrel it's a good idea to also have a lower configured for pistol use (meaning the buffer tube can't accept a stock). That's the way most people do it while they're waiting for their tax stamp.