You can argue about definitions and what or what is not a misnomer in our modern political context but Reagan very famously quoted himself “I believe the very heart and soul of conservatism is libertarianism.”
I think we concur that he who defines, wins.
I believe a credible argument could be made that the libertarianism represented by Freedom Communications and the Orange County Register, an early backer of Ronald Reagan, is very different from the amoral or even anti-morality libertarianism we're seeing today. The older libertarianism taught that there is such a thing as social morality that's necessary to govern people's personal lives to have a functioning society, but taught that it's the role of the family and the church to teach morality, not the role of the government.
I might be able to concur with that point of view in the 1950s or earlier when we still had a common moral consensus, but today we're seeing liberals who, when they have taken over government, actively use it to go to war against traditional morality. In our current context today I believe we must fight against libertarianism as a steppingstone toward liberalism, but I'm not sure I would have said that a few decades ago.
I used to work for Freedom Communications, BTW, and I'm not unaware of the influence of people like that in the history of the Republican Party in California and more generally in the West. Freedom Communications was in many ways the last significant conservative media operation in the United States after the Pulliams sold their newspaper company centered in Indiana and Arizona, and I've been disappointed to see what's happened to the company as the newspaper industry has collapsed.