Capitalism only means unconditional private property rights in the means of production. It doesn't require separation of powers or anything else, it can and has historically worked great with fascism.

"Money decisions" are also absolutely power decisions. The power concentration isn't even mainly in the decisions of the rich, but simply in that they are rich. You need to work building yachts rather than bake bread because the yacht buyers have so fabulously much more money to spare than the bread buyers. And the money they don't spend is even more powerful since it controls society's investments. Our society orbits around wealth, that's what capitalism is in practice.

There are famously capitalist countries that are monarchies (like the UK), so as written that has to be wrong. There isn't really a definition of what "capitalism" means, although we can be fairly confident it involves people owning capital. Arguably it means that the people who own the capital should control the lion's share of political power (which, coincidentally, is how the US works).

The thing that makes communism (and fascism to a lesser extent) anti-capitalist are that it rejects the idea people should be allowed to own and control capital. So there is a clear incompatibility with communism even if the exact nature of capitalism isn't clear.

that seems a bit naive. you can think of it like that, but then you'll find actual practicing capitalists (not capitalism, the generalized concept) will always cozy up to centers of coercive power, until they're the ones weilding those coercive powers. you've basically reinvented the medieval distinction between the church and the secular authority. the church would never burn people alive, they just determine if a person is spreading heresy and the secular authority will do what it will.

in reality it's all one system and it exists in reality. the system that exists in theory is just a model to carry around in your head.