The very first page defines it:
Eine Inzidenzstruktur $\mathfrak A$, die (A1), (A2) und (A3,h) für eine natürliche Zahl $h$ erfüllt, heißt $h$-affine Inzidenzebene. Sind $\mathfrak P, \mathfrak G$ endlich, so heiß $\mathfrak A$ endlich. Eine $2$-affine Ebene heißt auch biaffin
In other words:
A bi-affine (incidence) plane is a structure satisfiying
$(A1), (A2)$ and $(A3,2)$:
$$\begin{align*}
(A1)\qquad & \text{for each } P, Q\in \mathfrak P \ \exists!\ g\in\mathfrak G \text{ such that } g \text{ incides with both } P,Q \\
(A2)\qquad & \text{There are } P,Q,R \in \mathfrak P \text{ such that no } g\in\mathfrak G \text{ incides with all three}\\
(A3,2)\qquad & \forall (P,g)\in\mathfrak P\times \mathfrak G: \ 1\le\pi(P,g) \le 2
\end{align*}$$
Definition 6 defines a $w$-line by $O(g) = n$ and a $v$-line by $O(g) = N$(where $N$ is the order and $n$ is the suborder)
Which is given in eqn. (5) / Definition 2 by
$$N = \max_{g\in\mathfrak G} O(g), \qquad n=\min_{g\in\mathfrak G} O(G)$$
Definition 7 defines a Type I biaffine plane to be one such that every point has a $w$-line inciding it.
Definition 8 defines a Type II biaffine plane to be one such that every point has a $v$-line inciding it.
