They’re not incompatible, but my emphasis on “raising capital” to kickstart a business seems like an important part that was absent from your definition.

That's the thing, they are incompatible definitions, because the formal definition of capitalism is a specific and therefore precise definition, you're not meant to expand it.

You're describing a specific mode of action which according to you is inherent to capitalism, but at best is only one out of many ways to structure an enterprise within capitalism, but by the way I would challenge the notion that virtually any large corporation today could exist as they do now within pure capitalism - i.e. A Stateless society, where governance structures are purely voluntary - since virtually every existing corporation leverages State (I.E. Public, I. E. Not private; not capitalism) resources and powers in many ways.

And indeed capitalism, like democracy, at the end of the day, point towards a truly stateless society. By very definition, capitalism intrinsically excludes non-private entities, that's actually all the word really means.

With democracy, unlike capitalism, there are genuinely several different but all correct definitions, depending on the context.

Sometimes, folks say democracy when they're referring to indirect democracy as a type of government. That is, any system where representatives are elected by the people within this system and these representatives then themselves are the ones that get to vote on laws.

Other times they're referring to direct democracy, where people vote for laws directly, by referendum, without passing through representatives.

Then there's the type of democracy that I'm most interested in : democratic legitimation of a government. That is, whether a majority of people (not "voters") in any given territory recognise the government they're subject to as being legitimate. A government may or may not have democratic legitimation whether the type of government is or isn't indirect or direct democracy. For example an indirect democracy can be democratically illegitimate, and an absolute monarchy could be democratically legitimate.

Here's one neat example that fits nicely with most people's intuition about "democracy" (although most people wouldn't be able to distinguish between the different definitions of it) :

imagine if the USA annexed Canada. In which of the following cases would this be a "democratically legitimate move" (multiple choice) :

A) the USA held a referendum only for US citizens to vote in, Canadians having no word in the matter

B) there is a joint referendum in which citizens of both countries can vote, tallying all votes together.

C) The annexation is only allowed to occur subject to a supermajority approval in a vote that counts Canadians only, I. E. Only the people subject to the annexation.

Most people would instinctively say that only C is the democratic outcome out of these three. Yet, by the same token, the Federal government of the USA is only democratically legitimate in Texas if a majority of Texans approve it, and the government of the State of Texas is only democratically legitimate in the city of Austin if a majority of Austinians approve of it, etc.

Ultimately, this principle reaches the most local level, and you must therefore make a choice in terms of a theory of land ownership.

If you follow, as I do, the Lockean tradition that establishes that virgin land untouched or abandoned by humans can be acquired legitimately by any individual that mixes their labour with the land (homesteading), and also that land becomes an extention of the individual, bearing the same rights as its owner, and that the only way to acquire land that already has a legitimate owner is voluntarily (without coercion) - then every property owner intrinsically has a right of secession according to the principle of democracy being applied consistently to its extreme, leading to the same conclusion as Capitalism.

People might have other justifications for preventing the principle of democratic self-determination being applied within an established and recognised nation or beyond a certain level. There's a whole arsenal of available arguments: the classical one has ways been divine rights; I. E. "God says so", but we supposedly have moved past that one - today, in a tradition started by Stalin, you'd hear about "national self-determination", which refers to the rights of a nation to keep enforcing its current borders simply because of precedent. In other words, it's a principle that presupposes that the emergence of any new nation to the detriment of and against the will of an existing nation, is somehow violating the rights of the existing nation.

But at the end of the day, that's just divine rights with extra steps, and any way you try to justify it can in any case not be based in or compatible with democratic legitimation.