Are you confusing libertarianism with anarchism?
From what I can tell, there is 'little-l' libertarian, in which someone seeks to keep government small to prevent abuse of freedoms and rights by said government, and 'big-L' Libertarian, in which someone thinks that corporations working via free market principles can solve all/most societal problems and government just gets in the way.
I don't really buy the second, though I have some respect for the first.
At least in the USA, the breakdown between "big L Libertarian" and "little l libertarian" is usually positioned as whether or not somebody is a member of the Libertarian Party and/or registered to vote as Libertarian, versus simply holding to generally libertarian principles.
The distinction you're talking about, if I understand you correctly, sounds more like the distinction between "minarchist libertarians" and "anarchist libertarians" (or "anarcho capitalists"). Minarchists support some (very small) government for functions where it seems to make sense to share the responsibility communally (commonly cited examples are national defense, law enforcement, highway construction, etc.) where the more radical libertarians want to banish essentially all "government" (at least as we know it today).
Not necessarily. When you get to the most radical edges of libertarianism, you find the anarcho-capitalist types who are (depending on who you ask) a type of anarchist. Libertarians who hold to a strict adherence to the "NIF Principle"[1] are opposed to most of what passes for law enforcement in contemporary society since it largely involves initiation of force and isn't for self-defense.
which a lot of libertarians subscribe to.
