I have heard some different concepts of what "capitalism" is, ideas ranging from Marxists to An-Caps, and I have to say that (as with "communism") it does little good to focus on semantics. If you advocate for something, you best represent it by stating that idea in plain language, without such divisive terms.
Nonetheless, capitalism is an economic system based on private ownership and accumulation of property, especially the means of production. There are idealists with various stipulations and caveats for their own brands, ranging from mutualism to monarchism (yes, I've heard from all of these kinds). And even from trusted materialists and journalists, I have seen limitations such as excluding derivatives markets and other, non-production-based situations.
What the Federal Reserve does, and what companies, banks and corporations do in the US, is very much a form of capitalism. And yet the US does have the most regulated markets in the world - institutions dug in more than even UK ones - and even, in many ways, the closest thing to socialism the world has seen.
The Federal gov't, based on representation and legitimization (legally and politically), exercises its influence on the markets within specified authority. As an element of this, the Federal Reserve acts to insure the stability and value of the US dollar, as well as setting interest rate targets and (presumably) maintaining a specific level of employment.
The latter (maintaining employment) has been repeatedly drawn into question as a policy, even since the beginning of the Federal Reserve - which brings me to my main point: the Reserve, like other institutions in a capitalist (or private, for-profit) economy, acts within the bounds, and ultimately for the interests, of the power-possessing individuals and institutions. In capitalism, that is the owners of the means of production.If not, then it is a group of individuals who soon will own those means of production.
Capitalism is a model of economics which goes back to monarchist times - in fact, many of the very basic ideological tenants were enshrined in that context, among the Ludwig von Mises clique (Austrian school). Why are socialism and democracy considered antithetical to monarchism as the liberal threat to kingdoms in 19th century Europe? Because in populist systems, power is expected to seek legitimacy in the form of consent from those governed. Capitalism is the ubiquitous form of the propertarian societies, which allow unbalanced, illegitimate power, and hence, fit the undemocratic models of state.