All three of these political philosophies are oriented toward limiting the power of traditional institutions. They have come to differ, over time, as to whether government, economic and educational elites are seen as being among those traditional institutions.
In origin Liberalism was opposed to the control that religion and aristocracy had over European culture. It was oriented directly toward allowing individuals to act according to their own precepts, and not being driven by historical influences. It generally had a deep faith in free markets and rationalistic individualism.
Whigs are liberal in that sense. In the original English form they are anti-monarchist Parliamentarians. And in the U.S. form, they are basically a 'States-Rights party' opposed to Federal judicial and executive power. They oppose centralization of influence and encourage devolution of all power other than basic legislative decision making.
Libertarians are an extremist form of Whiggery that distrusts the centralization of even legislative power and favors extreme independence of thought and action with a minimum of government interference. Modern neo-Conservatism that wants "to starve the government down to the size where we can drown it in a bathtub" is more a flavor of this kind of abstract Liberalism than of traditional Conservatism.
But Socialism inflected the main line of Liberalism in the direction of considering the wealthy to be one of those controlling institutions, whether that wealth was aristocratically assigned, or earned by generations of industry. Labor movements led to modern Social Liberalism wherein the government protects the average citizen both from tradition and from the upper social classes.
The balance between these forces had different influences on the Continent and in the English-speaking world. Continental Liberalism remains largely rationalistic, favoring either complete equality or absolute individualism as the preferred position government should take toward everyone. It mixes the perspectives of Socialism and Liberalism.
The U.S. and Commonwealth version of Liberalism has, instead, tried to find a balance between the two, and in the process become extraordinarily complex. This has allowed liberalism to take on some of the empathic sentiments of the religious and the care-taking instincts of the educated elites that it was initially meant to purge from society. And this has allowed it to get behind a succession of Progressivist periods that have created and broadened institutions that try to indirectly redistribute wealth to take care of the poor and create social mobility, and that handle specific abuses (like the aftermath of slavery, or the movement of women beyond the home-front) with direct ad-hoc remedies that may limit individual freedom and defy rationalistic egalitarianism. But it has also resulted in what Libertarians often mock as 'the nanny state' in which the 'smart' people make decisions to protect us from each other -- and they see it as a direct betrayal of the origin and basis of the whole idea of Liberalism.
Given those trends, being an "Old Whig, with an emphasis on 'Old'", would mean that one adopts a form of Liberalism that avoids Socialist content and well-meaning elitist social experimentation, but is not Libertarian, and still supports a government strong enough to serve a given sense of national priorities.
