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Hi,
I have created a separate experimental version of the model, which may be found here. https://www.thingiverse.com/thing:2351153
I have only added circular hinge support to the experimental version for now. It has an option for a completely circular hinge, as well as an option where the knuckle joint is left uncut, but the leaves are cut into semicircles from the edge of the knuckle joint, roughly as you describe above.

You can find the "Hinge Shape" parameter in the "experimental-parameters" group in the Customizer.

For the sake of mathematical convenience, I used the existing "hinge-parameters / Leaf Height" parameter, to set the radius of the circular hinge shape, both for full circle and semicircle leaf options. In this case, the radius is half the value of "hinge-parameters / Leaf Height".

To configure the fastener layout, use the following existing fastener parameters.
"fastener-parameters / Fastener Count"
"fastener-parameters / Fastener Column Count"
"fastener-parameters / Fastener Margin Distance"

Experimental Parametric Hinge