The setting is the one period model of a financial market with $d+1$ assets, where the $0$-th asset is a riskless bond:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\overline\pi \in \mathbb{R}_+^{d+1}$ be an initial price vector with $\pi^0 := 1$. Let $\overline S:=(S^0,S^1,\dots,S^d)$ be the final price vector, where $S^0 \equiv 1+r, r>-1$ and $S^i, i \ge 1$ are non-negative RVs. At inital time, a portfolio $\overline\xi \in \mathbb{R}^{d+1}$ can be chosen. $\overline\pi \cdot \overline\xi$ is the initial price of the porfolio, $\overline\xi \cdot \overline S$ is the final value of the portfolio, where the dot denotes the standard inner product.
Definition: $\overline\xi \in \mathbb{R}^{d+1}$ is an arbitrage opportunity, if
$\quad$1. $\overline\pi \cdot \overline\xi \le 0$,
$\quad$2. $\mathbb{P}(\overline\xi \cdot \overline S \ge 0 ) = 1$ and
$\quad$3. $\mathbb{P}(\overline\xi \cdot \overline S > 0 ) > 0$.
The book summarizes the definition as follows:
Intuitively, an arbitrage opportunity is an investment strategy that yields with strictly positive probability a strictly positive profit and is not exposed to any downside risk.
But if I had to make this "intuitive" definition precise, I would rather drop the first requirement and replace $\overline\xi \cdot \overline S$ by $\overline\xi \cdot \overline S-\overline\xi \cdot \overline \pi$ in the second and third requirement. What is wrong with my understanding of the model?
This appears to be the authors way of saying
$$\overline S \cdot \overline\xi \ge \overline\pi \cdot \overline\xi$$
with an arbitrary $0$ point between them.