Classical liberalism, libertarianism, and anarchism are absolutely not interchangeable.
Liberalism is a group of related ideologies that descend from (or claim to draw from) Western Enlightenment thinkers. Classical liberals are people who claim to represent the views of those thinkers, in contrast with the modern generation of liberals who will pay lip service to modern notions of social welfare. Classical liberals typically advocate for free markets, low/no taxes especially for the wealthy, and "limited" government in the sense that it is only limited in their authority to help the poor and downtrodden [1].
Libertarianism is a term used for several movements and ideologies. Without qualification, I feel like it has little use. The original usage of the word was in relation to early libertarian leftism [4]. However, the current usage of the word is for """libertarian""" capitalists. Their ideology retains private property, but calls for the nominal abolition or reduction of government power.
Anarchism is, in my view, the label applying to those ideologies and movements that oppose all forms of authority and hierarchy. In my opinion, a consistent anarchist ought to oppose liberal democracy and liberal political systems in general. Liberals are creatures of the right because they don't oppose capitalism and usually don't oppose hierarchy on principle. Liberals have a historical tendency to betray movements that are too hostile towards their status quo, even if that means siding with obvious bigots who would kill them if given the chance. "Scratch a liberal, and a fascist bleeds." [2]
Some people use the term "libertarian socialist" (or "communist" depending on their economic beliefs) to denote those socialists (or communists) who believe in opposing authority and hierarchy, but believe that full abolition of the state is impractical. We otherwise draw from the same thinkers. Basically, all anarchist socialists [3] are libertarian socialists, but some libertarian socialists may not strictly be anarchists (but many are).
In my view, the error that "rank and file" liberals make is in taking the words of Enlightenment philosophers out of context (assuming they're arguing in good faith, which is usually a generous assumption). This causes them to inherit the biases of those philosophers, and this is especially true about classical liberals who distinctly seek out an outdated liberal perspective that doesn't challenge their bigotry. This "useful idiot" property of liberals is exploited by capitalists to create ideologues who apologize for capitalism, and by fascists who exploit liberals' aversion to change to oppose socialist movements.
Make no mistake: liberalism is the ideology of capital, and liberal writings from at least the previous century onward should be regarded as obvious capitalist propaganda. That being said, I'm not of the opinion that we need to dispose of all Enlightenment thought, but we must read into these ideals with a great deal of skepticism and context.
[1] This is not meant to imply that the State is even designed to help the poor, or that state welfare is the be-all end-all solution to poverty. However, as a stopgap measure, government welfare is better than literally nothing, although not equivalent to genuine mutual aid or building communities.
[2] I don't mean that liberals are all outright fascists. What that expression really says is that in times of distress, liberals will side with fascists before liberation movements. However, even in comfort, liberal perspectives are...kinda wack, because they're so obviously trying to justify oppression.
[3] For various reasons, including historical atrocities committed by self-identified socialists and arguments that socialist frameworks, even those that guarantee freedom, are themselves hierarchies that need to be dismantled, not every anarchist identifies as a socialist or even a leftist. Personally, I don't think that socialist frameworks are inherently hierarchical, but I understand that even supposedly "anarchist" societies have the possibility to develop hierarchies and injustice if we become complacent, and that we should be continuously re-evaluating our relationship with the rest of the Left.
[4] EDIT: replaced "communism" with "leftism", removed "e.g. direct transition to stateless, classless, moneyless society".