Nah. Rothbard is an anarcho-capitalist. He coined the phrase to describe himself. The Libertarian Party is most certainly an offshoot of these nuts. Ayn Rand is another devotee of this sort of anarchism.
I’m no more confused than Bill Buckley was (until he decided it was a good idea to ban tobacco). I would argue that what we call “libertarianism” is no more than a recovery of the 18th century idea of liberalism. Whether that be termed “minarchist” or some other nonsensical and hyper-modern label, the concept has been unchanged since before the American Revolution and most definitely before 1971.
If you don't want to be linked with the likes of the Paul's, Rothbard, Rockwell, DiLorenzo and others. I suggest you find a label that better suits your politics. Btw, classic liberal is not it.