Steiner conic

Remarks

The condition $\pi(UV) \ne UV$ makes $m$ and $\pi(m)$ distinct lines for every $m$, so each intersection is a single point. The points $U$ and $V$ lie on $C$: $UV \cap \pi(UV) = V$, and the line $m$ with $\pi(m) = UV$ gives $m \cap UV = U$.

Example: in the projective plane over a field $F$, let $U = [1 : 0 : 0]$ and $V = [0 : 0 : 1]$. The lines through $U$ are $by + cz = 0$ and the lines through $V$ are $ax + by = 0$, with $(b, c)$ and $(a, b)$ up to scaling. Let $\pi$ send $by + cz = 0$ to $bx + cy = 0$; this map is linear in the parameters, and over a field such maps are projectivities. The line $UV$ is $y = 0$, and $\pi$ sends it to $x = 0$, so $\pi(UV) \ne UV$. Solving $by + cz = 0$ and $bx + cy = 0$ with $t = -c/b$ gives the points $[t^2 : t : 1]$, and $b = 0$ gives $U$. So $C$ is the conic $xz = y^2$.

Non-example: let $a$ be a line through neither $U$ nor $V$, and let $\pi(m)$ be the line joining $V$ to $m \cap a$. Then $\pi(UV) = UV$, so the condition fails. For $m \ne UV$ the point $m \cap \pi(m)$ is $m \cap a$, so these points are the points of the line $a$ other than $a \cap UV$, not a conic.

In a plane over a field, Steiner's theorem says that the Steiner conics are exactly the non-degenerate projective conics.