The term itself, "classical liberal," is not made up by the right, but the context in which it is usually advanced -- a suggestion that modern liberals are not liberals, a focusing on a peripheral characteristic of classical liberalism instead of its core beliefs -- is a right-wing distortion of what classical liberalism was.
It was NOT small-government conservatism. It was NOT modern libertarianism, although some points do exist in common there.
Basically, classical liberalism was liberalism as it existed prior to the industrial revolution. Classical liberalism was liberalism appropriate to an agrarian economy still striving to pull itself out of monarchy and feudalism. (Both of which liberals opposed.)
The core value of liberalism, whether classical or modern, has always been liberty -- but the liberty of the common man, not the liberty of the elite, the rich, the powerful. Liberals have always sought to limit and contain the liberty of the elite, the rich, and the powerful so as to protect the liberty and the rights of the common man. In that, it has not changed from the days of John Locke, Thomas Jefferson and Adam Smith to modern times.
But in the days of John Locke, Thomas Jefferson and Adam Smith, the elite, the rich, and the powerful who could oppress the common man and threaten his liberty usually had hereditary titles and employed the government directly to aid in oppressing his victims. Hence, for example, Adam Smith's advocacy of free market economics: he saw that as an alternative to state-declared monopolies for the benefit of titled nobles, or chartered privileged positions like that of the East India Company. Compared to an economy in which the government enforced the privileges of the elite, the rich, and the powerful, a free market is a liberal dream. But today, having actually seen the effects of a free market and how it breeds its own form of elite, rich, and powerful people who are capable of oppressing the common man without even resorting to government power (not that they've ever been shy about resorting to government power, either), liberals have come to see it in a less rosy light, and to recognize that some modification of the ideal might just be called for. Adam Smith, were he transported to modern times and given a while to catch up on history, would not disagree. If he were alive to write The Wealth of Nations, 21st Century Edition, it would incorporate a lot of socialist-seeming elements that were not in the original. His values were the same as those held by modern liberals, even if his specific ideas, in the specific context in which he wrote them, would not necessarily work in service to those values today.
In today's world, we know that the government is a danger to liberty, and that there are still times when direct government infringement of the people's rights is the clear and present danger that must be fought. Modern liberals like classical liberals insist that government be accountable to the people, governed by law, kept safe through checks and balances, and restrained from committing egregious violations against the common man, such as depriving accused criminals of due process, or violating free speech or the right of privacy, or attempting to establish a government-sponsored religion. In this respect, liberals are now and always have been advocates of limited government.
But in today's world, we also know, having learned the hard way, that government is not the ONLY danger to liberty, and that private agencies -- the elite, the rich, and the powerful, and the corporations that are the principle institutions (other than government) through which they work their oppression -- can be as deadly to the liberty of the common man as the state, or nearly so, and the state is the only institution that we can use to restrain them. And so in that context, where state power is needed to restrain powerful non-governmental threats to liberty, liberals have become advocates of big government. Also, in today's world, we also know that there are public services that can be provided only by government which serve to empower the common man -- relieving the dangers of unemployment, providing medical care that he could not afford on his own, providing free education, and providing information and other resources -- and hence aid liberty, and in this context, too, modern liberals have become advocates of more government that classical liberals tended to advocate. Although conceivably that last was only because they lived in poorer times that could not afford those services, or else the technology didn't exist; let's remember that Thomas Jefferson was a huge advocate of public schooling.
So there it is. Yes, there are differences between modern and classical liberalism but they are trivial. (Except, perhaps, to those who are obsessed with the size and scope of government, who confuse an absence of government with a presence of liberty. It's been truly said that to a man with a hammer, all things look like nails.) In terms of core values, liberalism remains the same now as it was then, and if it has adapted to the times we live in rather than remaining fixated on methods that were appropriate to a time that no longer exists, that only means that we have not been consistently stupid.
And yes, I would certainly vote for a liberal for president. Now if only one was running . . .