Hi, minarchist here. You are almost right in the distinction between right and left anarchism being about private property. But it's a little more profound. The main difference radicates in the conception of legitimate power. For a right anarchist, power is illegitimate only when it's carried out through violence. Meanwhile, for a left anarchist almost any power hierarchy is illegitimate in itself. To the extent that private property could give rise to power hierarchies, a left anarchist will reject it. But it's only a particular case derived from the core belief.

> "anarcho-capitalist" is inherently paradoxical

Well, anarcho-capitalists believe that there are ways to provide property rights outside the state. As a minarchist, I'm skeptical about it, but if you believe it there's no paradox.

On the other hand, I do find deeply paradoxical the left anarchist stance on rejecting capitalism but also (AFAIK) reject any means to stop its emergence in society.

> There is also another conversation to be had about minarchist/libertarian ideas basically wanting to remove all the good bits of the state [...] but keeping the bad parts

As a minarchist, I strive for reducing the weight of the state as much as humanly possible. Justice, defense, and a minimal social security net are the only services I can't think how to provide without the state. If you think these are the bad parts I'd love to hear a viable alternative, maybe you can make me fully anarcho-capitalist ;).