It is true that if one has the position vector as a function of time, this is all we need to analyze the motion. Basically, the evolution function of the position of one point ${\bf r}(t)$ contains the information about the space curve corresponding to all the instantaneous positions in the parametric form, where time acts as the parameter.
However, in some cases, one can be interested only in the space curve without information about time. In such a case, any change in the parametrization of the curve can be used (${\bf r}(s)$, where the parameter $s$ depends on time $t$ in a one-to-one way). If necessary, one could introduce different terms for denoting ${\bf r}(t)$ (time evolution) or the parametric curve ${\bf r}(s)$, trajectory, although usually, in the English textbooks, both are referred as trajectories.
In some special cases, in two dimensions, it is possible to invert the functional dependence of one coordinate on the parameter to express the curve in a nonparametric way, for example, as $y=y(x)$. Such cases make clear the information loss corresponding to eliminating the time dependence.
For example, we can describe the evolution of the position of a point moving in a uniform gravitational field, starting from the origin, as
$$
\begin{align}
x(t) &= v_{x,0} t \\
y(t) &= v_{y,0} t - \frac{g}{2}t^2
\end{align}
$$
but we can describe this time-parametrized curve using other parameters, non-linearly related to time, or in a non-parametric way as
$$
y(x)=\frac{v_{y,0} }{v_{x,0} } x - \frac{g}{2v_{x,0}^2} x^2.
$$
The previous formula is what your textbook defines as trajectory equation.
