Is this axiomatization of affine plane categorical?

Question

First I'll give some definitions.

Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms:

1. For any two distinct points $a,b$ there exists exactly one line $L$ such that $a,b\in L$.
2. For any line $L$ there exist two distinct points $a,b$ such that $a,b\in L$.
3. There exist three distinct points not lying on one line.

Next let $B_{\mathbb{R}}\subset\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ be standard (strict) betweenness relation on $\mathbb{R}$ i.e. $$B_{\mathbb{R}}(abc):\iff\left(a<b<c \vee c<b<a\right)$$

My definition of affine plane is as follows:

We say that $(P,\mathcal{L},B)$ is an affine plane whenever

1. $(P,\mathcal{L})$ is a model of incidence axioms.
2. $B(abc)$ implies that $a,b,c$ are collinear.
3. For any line $L$: $(L,B|_{L\times L\times L})$ is isomorphic to $(\mathbb{R},B_{\mathbb{R}})$.
4. Pasch's axioms holds.
5. For any line $L$ and point $p\notin L$ there is exactly one line parallel to $L$ passing through $p$.

Question:

Is this definition standard or equivalent to standard definitions of affine plane in terms of lines and ternary betweenness relation? And what is more important to me: Is this axiomatization categorical?

I had an idea to prove that my affine plane is isomorphic to $\mathbb{R}^2$. The idea was to pick two intersecting lines $L_1,L_2$ and isomorphisms $\xi_1:L_1\rightarrow\mathbb{R},\xi_2:L_2\rightarrow\mathbb{R}$ and to map any point $p$ on a plane to $(\xi_1(\pi_1(p)),\xi_2(\pi_2(p)))\in\mathbb{R}^2$ where $\pi_1,\pi_2$ are parallel projections to $L_1,L_2$. I think I know how to prove that this mapping is a bijection, yet I don't know how to prove it preserves betweenness. Not even sure it does.