To be clear, I was not the person you were responding to, simply someone describing a relatively old set of philosophies.
We're talking about the classification of a 170-year-old journal that operates out of the UK. I don't think the suggestion that libertarianism (particularly American libertarianism) is largely irrelevant when discussing The Economist is surprising, or shocking, or even terribly controversial. The suggestion that it would start a flame war is particularly odd as far as the content of the suggestion was concerned, although the tone of the comment you were responding to was a little blustery.
I'm sure a lot of the staff of The Economist and a lot of libertarians hold, say, Milton Friedman in high regard.
The clear cut distinction between classical liberalism and libertarianism, however, is not merely my own personal opinion. Although the intellectual landscape is complex and variegated, there's a fundamental difference in how the two lines of thinking argue, and what they accept as first principals. Classical liberalism is an institutionalist view that derives policy decisions using a utilitarian set of values that balances rights against other measures of well-being and sustainability.
Libertarianism, conversely, uses a first-principals rights-based way of approaching decisions. Just as an example, here's the front page of the Libertarian Party in the United States:
"<The Libertarian Party> is the only political party that respects you as a unique and responsible individual. Our slogan is that we are 'The Party of Principle' because we stand firmly on our principles. Libertarians strongly oppose any government interference in your personal, family, and business decisions. Essentially, we believe all Americans should be free to live their lives and pursue their interests as they see fit as long as they do no harm to another."
This is pretty close to antithetical to classical liberalism. Classical liberalism explicitly seeks to suborn individual rights (which are not enshrined as intrinsically valuable) in a way that doesn't make us worse off than we started. The minimalism is classical liberalism is about outcomes, not about rights.
Obviously that's just a cheap banner page and not representative of libertarianism to everyone. I was surprised to find that it was closer to the original French meaning than I expected (which isn't important, but might be interesting).