For example the term minimal statism is a bit clunky, libertarians use the term minarchism.
Also I don't care for this phrase (removed since): its (government's) function is only to keep people from harming each other. I don't think that really captures it. If you own a bookstore and I open a competing one nextdoor I've hurt you. If a husband cheats on his wife he hurts her. To libertarians the only purpose of government is to protect the rights of its citizens(those rights being defined by the non-aggression axiom and its corallaries).
Fare: indeed, to libertarians, the criterion is not whether someone feels hurt, but whether his rights have been respected or not. "hurting" or "profiting" is always respective to other imaginary worlds - and you can always imagine a world where you're better off or worse off. The only legitimate comparison when demanding justice, for libertarians, is between the world where your right has been respected, and the world where it wasn't.
MemoryHole.com: Yes libertarians(of the minarchist? persuasion) support state protection of property rights. But the DMCA brings up questions of intellectual property rights and there is wide disagreement about that among libertarians.
Fare Most libertarians think that property rights are natural, and that legislation cannot or must not arbitrarily create or destroy them, that it can only discover and claim those rights that exist out of natural law. Actually, many libertarians (particularly the anarchist ones) believe that legislation should not exist at all, and/or should not be a state monopoly, but be done by competing law agencies that people freely adhere to or not.
Fare Some libertarians believe that intellectual property is a natural right, some utilitarian libertarians think that it is an acceptable governmental privilege. The most radical libertarians, anarcho-capitalists, resent state protection of property, and reject intellectual property, at least as we know it.
Although Libertarians look to classical liberals as their origins, I think its wrong to view their views as identical. Though on many issues (e.g. dislike for large or powerful governments) classical liberals and libertarians are similar, I doubt that classical liberals had the same views on all issues as modern libertarians do. (For example, many modern libertarians seem to have an almost Randian enthusiasm for capitalism; Adam Smith, IIRC, saw it not so much as a good thing as the least of the possible evils.)
Furthermore, I think a lot of liberals (in the recent American sense of the word) see their origins in classical liberalism also. (And I think the historical origin of modern liberalism lies in part in classical liberalism as well.) They would argue that where they differ from classical liberalism, they are not so much in opposition to it as a natural development of it. -- Simon J Kissane
Besides, no non-libertarian ideology claims classical liberalism as its root, except in as much as they consider it part of the universal tradition, and after rejecting half of the classical liberal claims. Such is the case for modern US "liberals" -- they are actually socialists who prefer Marx to Smith.
But since I see room for dissensions here, I suppose there should be a specific article about classical liberalism, and another one on libertarianism.
Your use of the word "ideology" is telling; political theories are not ideologies.
The assertion "Besides, no libertarian ideology claims classical liberalism as its root, except in as much as they consider it part of the universal tradition, and after rejecting half of the classical liberal claims" is as valid as the one you made.
Libertarianism as you define simply can't be a synonym for classical liberalism; libertarianism is a current, 20th-century political theory defined by 20th-century theorists; classical liberalism a historical term for a political theory specific to an earlier era. Asserting that libertarianism and classical liberalism are the same is like asserting that neo-classical and ancient Greek architecture are the same. It's reconfiguring the past into current expectations and understandings.
The past and the present, in reference to social phenomena, cannot be equivalent. Those who believe that they are are idealists or ideologists.
I'd buy that classic liberalism is a precursor to libertarianism.