Confusion with what the metric gives when mapping a surface to the complex plane

$\newcommand{\Cpx}{\mathbf{C}}$The formula $$ d\hat{s} = \lambda(z, \gamma)\, dz $$ must be interpreted not as equating a spatial displacement with a scalar multiple of the "flat" displacement $dz$, but as an equality between an infinitesimal displacement $d\hat{s} = \hat{q} - \hat{z}$ in the tangent plane to the hemisphere at $\hat{z}$ and the image of the "flat" displacement $dz = q - z$ under the differential of (the inverse of) central projection from the plane to the hemisphere.

More precisely, if $F:\Cpx \to H$ denotes the inverse projection from the plane to the open hemisphere, then $\hat{z} = F(z)$ and (speaking infinitesimally) $$ \hat{z} + d\hat{s} = \hat{q} = F(q) = F(z + dz) = F(z) + (DF(z))(dz); $$ consequently, $d\hat{s} = (DF(z))(dz)$.

(The identification of $(DF(z))(dz)$ with a scalar multiple of $dz$ presumably represents the differential with respect to some basis.)