Liberal is supposed to mean “freedom.” When a government tells you what to do, that isn’t liberal. Of course there is a government and even classical liberals acknowledge the need for some government, but liberal really should mean a support of a government that provides for a common defense and has a court system that would provide for the enforcement of civil contracts. A “liberal” government doesn’t mandate actions of private entities unless those mandates are to protect the life, liberty or property of other individuals.

Mandating gender on corporate boards doesn’t protect any of that. Nobody has a right to serve on a board and not serving on a board isn’t threatening anyone’s right to pursue happiness or live their life as they see fit. In fact, it does the opposite: it imposes a requirement that limits the freedom of people to run their business how they see fit. Unless the running of that business infringes on the rights of others, the government really has no business with that sort of mandate. And the key to understanding this is that no person has a “right” to success or a right to a specific job — they have a right to pursue happiness, but not a right to actually be happy.

Progressivism is well-intentioned totalitarianism. There is nothing wrong with supporting progressivism, but let’s not pretend is has anything to do with liberalism.

Not the OP, but the best way to contrast liberalism is from the perspective of Puritanism. Puritans were certainly not liberal socially, but they were also not liberal governmentally - many of the laws in Puritanical societies were highly restrictive, and pre-modern liberalism had its roots in creating a more free society - Puritanism is a non-negligible part of American history.

Modern leftism tends to be more authoritarian - it encourages restrictive legislation as a mechanism for progress. This is the dystopian future alluded to in a lot of e.g. Heinlein's work - things like forced sexual liberalism, erosion of parental rights or even parentage itself, elimination of the individual for the cause, and discrimination against the individual based on factors outside of their control.

Americans tend to view political affiliation as a binary - left or right, progressive or conservative, liberal or authoritarian. That's because, due to our history, we have always tended to have a leftist, progressive, liberal party contrasted with a right-leaning, conservative, authoritarian party.

That language is in and of itself somewhat complex - prohibitionists were considered progressives in their time, though today we would view them as otherwise.

In many other countries, the right-wing party is called "Liberal". The reason is that the history of liberalism is closely intertwined with the history of economics, but they started to diverge when economists started to argue that wage floors and welfare could result in slower economic progress. The most extreme example of the latter argument is John von Neumann's turnpike theory:

http://en.wikipedia.org/wiki/Turnpike_theory

There was a sort of "split" between people who wanted to maximize economic growth and people who wanted to optimize for some other measures of social well-being, but, crucially, they were both operating within the "social metric optimization" context of 19th century liberalism, aka classical liberalism, which included e.g. Bentham, Mill, and Schopenhauer. The major 19th century competitors to this philosophy were the various flavors of socialism, largely comprising the followers of Marx and Ricardo, which prefer to build social structures that naturally encourage human freedom (at least in theory), and also the various flavors of conservatism, which aim to preserve historical social structures believed to be beneficial for reasons that are often not clearly expressed. In countries subject to colonial conquest -- that is, everywhere except Western Europe, Russia, Turkey, Iran, Oman, Thailand, China and Japan -- these historical social structures were destroyed, and along with them the population's appetite for conservatism. In Japan, Italy and Germany, conservatism was heavily curtailed after World War II.

When liberalism's main competitor was conservatism, the growth-maximizers tended to align themselves with the conservatives and liberals were on the "left". But when it was socialism, the well-being-maximizers aligned themselves with the socialists and liberals were on the "right". What connects the two kinds of liberals is a reliance on numerical measurements of individual status (wealth, happiness, lifespan) and dynamical theories (economics, sociology) to define, promote and produce social progress.

Small "d" democratic is something done through the process of democracy. Capital "D" Democratic is something belonging to the Democrats. It's that capitalization which indicates that it's a proper noun and referring to the party with that name. Without the capitalization it's not a proper noun and does not refer to the party.

I capitalized Democratic for a reason. I'm talking about the Democratic Party specifically.