[mindcrime]

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.

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).