Dear Olivier, in line with the more advanced nature of this site, let me give an example of a less elementary nature.

Consider a compact Riemann surface $X$ of genus 2 and on it stable vector bundles $E$ of rank 2 whose determinant bundle $\Lambda ^2E$ is isomorphic to some fixed line bundle $L$ of degree $-1$. Newstead has proved that the moduli space of those vector bundles is the intersection of two quadrics in five-dimesional projective space $\mathbb P^5(\mathbb C)$. And one of those quadrics is the Klein quadric in $\mathbb P^5(\mathbb C)$ parametrizing the lines in some three-dimensional projective space canonically associated to $X$ and $L$. (A Klein quadric is the quadric you mention in number (v) of your list.)

References
P E. Newstead Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215.
For a geometric description including the role of the Klein quadric, see:
M. S. Narasimhan and S. Ramanan Moduli of Vector Bundles on a Compact Riemann Surface, Annals of Mathematics, Vol. 89, No. 1, 1969 , pp. 14-51.