Excellent question. Modern scholarship is contemptuous of left, right and center political spectrum; but it remains very important as the most widely used metric for describing modern and historical leaders and philosophies in use.
Your question presupposes that there is ambiguity in equating classic liberalism which, as you state, is better today classified as a moderate right leaning ideology(libertarianism); than with progressivism, the left, or modern liberalism(all synonyms). What you are missing is the understanding of what distinguishes left and right along with historical perspective.
The left is about new ideas, new solution when problems occur. Classic liberalism in the Age of Enlightenment when Jefferson, Paine, Beccaria, Baruch Spinoza, Diderot, Kant, Hume, Rousseau and Adam Smith first expressed them were new and thus liberal. Just as we say Stalin and Lenin were extreme examples of left ideology because they set out to create a new form of government.
The right or conservatism is about preserving the status quo. When conservatives face a problem they try previously used solutions or failing that reforming the existing system. We call the Roman senator Cato a conservative because at the end of the Roman Republic he dedicated himself to preserving the republic, the old system. He opposed the new, opposed empire. Alexander Hamilton is said to be the father of American conservatism for taking the opposite position. Hamilton favored recreating a monarchy in an age when monarchies were the norm. Hamilton and Cato both believed in the tried and true forms of government for there times but due to there periods supported opposite systems.
Hitler and Mussolini(fascism) are extreme examples of conservatism because in both Germany and Italy they were about recreating what had existed in the past. Hitler wanted to create the third Reich. Mussolini wanted to recreate the Roman Empire.
Each side has its advantages and disadvantages for moderate subscribers. Right leaning or conservatism tend to make safe well understood changes when confronting problems; however also by definition these solutions are compromises as there typically are reasons previous policies where changed. Conservatism really falls down when society faces systemic problems history has no answer for and reform is insufficient to address. ( ending slavery, labor laws after the dawn of the Industrial Age, trust busting, civil rights, woman’s suffrage ) Still conservative solutions are generally good safe solutions.
Left leaning or liberal solutions are inherently more dangerous with more unforeseen consequences. New ideas are inherently more risky than well understood previously tried solutions. Still without new ideas there is no progress. Most old ideas where at one time new ideas. Which brings us back along to libertarianism or classic liberalism.
Typically conservative ideology is most attractive after prolonged liberal rule; when society wants to introduce normalcy or just get things working again. Not coincidentally liberal ideology is most attractive and appealing after prolonged conservative rule. When society is struggling with systemic problems which history has no answer for.