Generalized quadrangle

Remarks

Counting gives $(s + 1)(st + 1)$ points and $(t + 1)(st + 1)$ lines. A generalized quadrangle is the same as a generalized polygon with $n = 4$. The finite polar spaces of rank $2$ are generalized quadrangles.

Example: the $(s + 1) \times (s + 1)$ grid, with the rows and columns as lines, has order $(s, 1)$. Clauses 1 and 2 are clear. For clause 3, let $p$ be off the row $\ell$. The column through $p$ meets $\ell$ in one point $q$, and the row through $p$ is disjoint from $\ell$, so exactly one line through $p$ meets $\ell$. The same check works with rows and columns exchanged.

Non-example: in a projective plane of order $n \ge 2$, every line through a point $p$ off $\ell$ meets $\ell$, so $n + 1 > 1$ lines through $p$ meet $\ell$ and clause 3 fails.

The symplectic quadrangle $W(q)$, formed by the points of $\mathrm{PG}(3, q)$ and the totally isotropic lines of a symplectic form, has order $(q, q)$.