I'm not too sure about the Liberal Alliance, but I think we can distinguish between the classical liberal parties of Europe like FDP and US-style Libertarians. The former are more establishment friendly pro-business parties who drain their support from the upper-middle class; while the Paul movement (as I see it) is much younger, disparate and anti-establishmentary.
Yes, but the actual policy content is mostly quite similar if you weed out the many parties (and factions of parties) that are de facto just Conservatives in anything but name. Moderate Libertarian and Classical Liberal seems to cover identical ideologies. It is just that Libertarianism is connected with being radical, so a lot of people seem reluctant to self identify as a Moderate Libertarian.
I still think there is a tangible difference between the two labels though. Like the Classical liberal says "I must lower taxes to encourage business", while the libertarian says "I must cut taxes,
because I am morally obligated to do so."