In https://encyclopediaofmath.org/wiki/Rational_number a rational number is defined as an equivalence class of pairs in the usual manner, and a rational number $r$ is called positive (negative) if it contains a rational fraction $\frac{a}{b}$ with $a$ and $b$ of the same sign (of different signs).
In https://www.math.wustl.edu/~freiwald/310rationals.pdf instead a nonzero rational number $\frac{a}{b}$ is called positive if it is possible to write $\frac{a}{b} = \frac{c}{d}$, where $c,d \in \mathbb{N}$, and $\frac{a}{b}$ is called negative if it is possible to write $\frac{a}{b} = \frac{c}{d}$, where exactly either $c \in \mathbb{N}$ or $d \in \mathbb{N}$ and not both.
Do the different notions of negative and positive rational numbers defined above coincide?