Anarchism and libertarianism originated in Europe in the 18th century as a branch of Socialism. What Americans call libertarianism is what the rest of the world calls anarcho-capitalism, a rather different ideology.
Anyway, Rothbard, a prominent anarcho-capitalist, states quite clearly that "libertarians" (i.e. anarcho-capitalists) are opposed to government, which contradicts your statement that anarcho-capitalists want "a strong government".
US libertarian party is based on small government but it very much wants a strong military. Some members are anarchists but the party is far more diverse than just that. So, referring to libertarianism in the US as identical to anarcho-capitalism isn’t accurate.
Anyway, I agree that Rothbard doesn’t want government, but I am saying he does want the rule of law which is a separation point from pure anarchism. How exactly you get a court system that’s not backed up by the threat of force without people simply ignoring it is a point of contention.
https://en.wikipedia.org/wiki/Libertarianism
“Libertarianism originated as a form of left-wing politics such as anti-authoritarian and anti-state socialists like anarchists,[6] especially social anarchists,[7] but more generally libertarian communists/Marxists and libertarian socialists.[8][9] These libertarians seek to abolish capitalism and private ownership of the means of production, or else to restrict their purview or effects to usufruct property norms, in favor of common or cooperative ownership and management, viewing private property as a barrier to freedom and liberty.”
That said, anarcho-capitalist still wants the rule of law they just don’t want to call it government. Rothbard's anarcho-capitalist society would operate under a mutually agreed-upon "legal code which would be generally accepted, and which the courts would pledge themselves to follow".[11] This legal code would recognize contracts, private property, self-ownership and tort law in keeping with the non-aggression principle.[11][12]