I would suggest you try not to think in labels and embrace the reality: no person is a democrat/republican/tory/labor/whatever. Those are political parties, sure, but what happens when a person disagrees with the platform of that party? Are they no longer a 'whatever'? No, they were never a part of the party to begin with, they just identified with the group that most closely aligned with their views.
Of course there are no pure neoliberal politicians as such a being would have no policies except cutting popular things and privatizing them. Hard to get elected. So we have neoliberalism with left or right characteristics as the mainstream ideologies.
The more modern way (still in Europe) is seen as reasonably minimal level of government regulation, but strongly applied. For example the libertarians I know are for minimal government (and associated taxes), but very strong consumer protections - this is because individuals have very little power against huge corporations, so it is just leveling the field.
But I believe that the government must protect the free market against capitalism's inherent tendency towards monopoly, and that likewise, it must protect consumers against exploitative business practices (which arguably, a properly competitive free market could help achieve).
Libertarians have been increasingly confused with anarchists and specifically anarcho-capitalists, neither of which believe in government.
While it has been humorous to see these faux-libertarians say in one breath they want small government, but in the very next breath demand to be saved, tankies and other far-left extremes have used this as an opportunity to lump all libertarians into the same boat.
There's no such thing as a "true" libertarian as most libertarians believe in small government, but what that government can do, is generally up to interpretation in all but the most extreme cases. A major central belief the non-aggression principle. However social issues tend to be more wishy-washy.
Prior to the balkanization of America Libertarian-lite could probably be approximated by a classical liberal.