You're right, these definitions aren't equivalent. Below I'll give a counterexample which has a multiplicative identity (for me rings always have multiplicative identity).
Let me rephrase both definitions in a way that I personally find easier to think about. A strict ordered ring (your first definition) is a ring $R$ equipped with a subset $R_{+}$, the positive elements of $R$, satisfying the following axioms:
- $R_{+}$ is disjoint from $R_{-} = - R_{+}$, neither of them contains $0$, and $R$ is the disjoint union $R_{+} \sqcup \{ 0 \} \sqcup R_{-}$ (trichotomy).
- $R_{+}$ is closed under addition (transitivity).
- $R_{+}$ is closed under multiplication.
(We recover the total order from here by defining $x < y$ iff $y - x \in R_{+}$.)
An ordered ring (Wikipedia's definition) is a ring $R$ equipped with a subset $R_{\ge 0}$, the nonnegative elements of $R$, satisfying the following axioms:
- The intersection of $R_{\ge 0}$ and $R_{\le 0} = - R_{\ge 0}$ is exactly $\{ 0 \}$, and $R = R_{\ge 0} \cup R_{\le 0}$.
- $R_{\ge 0}$ is closed under addition (transitivity).
- $R_{\ge 0}$ is closed under multiplication.
Which of these definitions seems more natural to you will depend on whether you find it more natural to think about total orders in terms of $<$ or in terms of $\le$. The first two axioms turn out to be equivalent (once we translate between "positive" and "nonnegative" by adding resp. removing $0$), but in the third axiom the difference is that the second definition allows the product of two positive elements to be zero, and so allows zero divisors, while the first definition implies that $R$ is an integral domain.
Here's an example of a non-strict ordered ring that seems pretty natural to me: consider the ring $R = \mathbb{R}[\varepsilon]/\varepsilon^2$, equipped with the ordering in which the nonnegative elements are $x + y \varepsilon$ such that either $x > 0$ or $x = 0$ and $y \ge 0$. Then $\varepsilon$ is positive but $\varepsilon^2 = 0$. This ring is a natural setting for comparing the growth rates of, say, two polynomials near a point.