I don't think it's bickering, so much as one of the recurrent problems in political science: it's full of technical jargon which is often at odds with the "popular" use of the same words.
Classical liberalism espoused democracy, the power of laissez-faire capitalism, and called for destratification of society. Of course, it frequently did so while only looking at white males with enough wealth to have any kind of interest in politics to begin with. This was contrasted with the conservatism of the period (18th/19th centuries), which argued for the retention of monarchy and the landed gentry as the arbiters of power in society.
The problem, according to Marx and others, was that while the liberal democracy was more free than serfdom, it was still inherently an unfree system because you merely traded dominance of a ruling class based on heredity to a ruling class based on wealth, rather the sort of egalitarian democracy that it hoped and claimed to be. I think you don't have to look around too hard today to admit that Marx had a pretty good argument. He also argued that capitalist democracy was a necessary stage in the progression of a society, and that socialism would arise once liberal democracy had burned itself out and created such a class inequity that it was not all that different from the monarchies it overthrew. This would lead to the "rise of the workers' proletariat", etc. (Lenin's problem was that he wanted to accelerate the timeline and skip the liberal democracy stage altogether.)
Conservatism in the strictly political science sense, isn't really tied to democracy or capitalism as core values. What we have in the US are two "classic" liberal parties, one which is socially conservative, and one which is socially moderate-to-liberal (by US standards). The former prefers "classical economics", which is to say supply-side economics; while the latter prefers Keynesian economics, particular during economic downturns. But as time marches on, it's difficult to say what a truly "conservative" party in the US would be. We never had dominance by a landed gentry or the Church the way that many European countries had, so there's no core segment there to resist change. Instead, we have people who have reconstructed an America-That-Never-Was: where brave, white Christian (typically Protestant) males conquered a savage wilderness and fought off all their enemies and made prosperous, moral livelihoods for themselves. And that's what they want to "return" to.
But because the actual history of the US (and unfortunately, the version I mock above is more or less the version that gets taught to most of our children) is considerably more colorful, nuanced and amoral/immoral....there's nowhere to return to. And if you dare to point that out (or to suggest that perhaps the best way for a country is to look forwards, rather than trying to regress 250 years), you're a goddamn pinko commie socialist f*g who hates America.
So yeah. That's why politics in this country can be so effed-up that we have a majority of Republican candidates denying evolution and global warming and thumping their Bible all the way to the ballot box. Because they think (and more importantly, a sizable portion of their constituents think) that that's what ol' George Washington and Thomas Jefferson would have wanted.
Ben Franklin, meanwhile, would have bitchslapped these people into next Tuesday. (Although he might have had a soft spot for Gingrich and Cain...Franklin liked the ladies too.)