I agree about the textbook author's style being confusing.
An NPN BJT saturates when the collector voltage goes below the base voltage (with respect to emitter) but not quite down to the emitter voltage.
At that point, the collector current is limited by the supply voltage and load resistance, and increasing the base current cannot increase it further.
However, if you reduce the load resistance, then you can achieve a higher collector current by increasing the base current.
To ensure saturation (which results in relatively low collector power dissipation, a good thing), you must make sure that the base current is more than sufficient, and the "DC beta" hFE = IC/IB, the simple ratio of DC currents (not the derivative) is the relevant parameter and the manufacturer should specify a minimum value so that the designer knows how much base current is required to guarantee saturation.