You're not wrong about your definition, but you're not right. Both the republican party and the democratic party observe the core principles and philosophy of their party respectively, to some degree, but the beliefs of what is considered conservatism, and what's considered liberalism varies from politician to politician, and from person to person, often times contradicting the core beliefs of each establishment. That's why I gave an all encompassing, general definition of each party, instead of getting into the nitty gritty with specific policies and politicians.
For example, most Republicans do want small government. Fiscal conservatives actually work toward that end by cutting taxes and bureaucratic spending. Same with the Tea Party and Libertarian republicans. On the other hand, Liberal Republicans are generally in support of larger government, and Traditional Republicans push for imposing their own social views through laws and legislation. And even between those 4 types of conservatives, there are varying degrees of conservatism that respectively lies between them. So you're not wrong when you say republican's don't really want small government, but you're not right either.
Here's something I've been sharing with a few of my conservative friends. Rand Paul basically calls out the republican senate on their bullshit on their repeal on Obamacare on a plan that will set the U.S. nearly 10 trillion more dollars into debt. (Sidenote: I was really pushing for Rand Paul to be nominated as the republican candidate, but it never came to fruition).
Also, Obamacare isn't privatized, it's subsidized through private companies. People were pretty much forced to by health insurance, which increased the cost of premiums, causing the price of health insurance to sky rocket. Since people who couldn't afford health insurance couldn't be denied by the insurance companies, everyone else had to pay the price. It was a forced subsidy that got way too expensive. I hold more libertarian views when it comes to government programs and taxes, so I feel like democrats are hypocritical when they claim they're for liberty and freedom, while at the same time mandate taxes through force. That's more of a personal view, so I don't hold it against the entire establishment.
You already know how I feel about socialism/communism, so you know how I feel about Bernie Sanders, but I agree that both establishments fail to represent the will of the people, or at least to an effective degree, and I can admit Bernie Sander's did a good job at listening to the general public's problems and acting accordingly.
(Ughh, why tf do my replies always end up so long).