Alex, go here for the simple non-linear DC model developed in 1954: Ebers & Moll, "Large-Signal Behavior of Junction Transistors".
That model (or all three of the equivalent models shown at that link) does not include the Early Effect (or the Late Effect that was integrated into the Gummel-Poon model in 1970.)
Ignoring the Early Effect impact and when the BC junction is reverse biased, the bipolar is in active mode and the collector acts similarly to a current source/sink.
When the BC junction goes into forward bias, then according to the Ebers-Moll model that BC junction diode follows the exponential behavior of the Shockley diode model.
At sufficiently low forward-biased BC voltages, the effect is negligible and the bipolar may still appear to be in active mode (the \$\beta\$ is still close to what is expected in active mode.) The collector is still acting mostly like a current source/sink because the impact of the forward biased BC junction is still sufficiently negligible. But it may be better to start thinking about this case as "very soft saturation" so that you are keeping tabs of the fact that the bipolar is moving towards hard saturation and where the collector's behavior will be quite different.
As the forward-biased BC voltages increase, the BC diode current contribution increases exponentially and eventually becomes quite substantial. Enough to seriously degrade \$\beta\$ and to cause the collector to behave more as a voltage source (more as a slight voltage difference between the emitter and collector.)
For small signal silicon bipolars operating at room temperature, a forward-based BC junction with less than \$\vert 400\:\text{mV}\vert\$ might still be considered in active mode. But it would not be wise, except when absolutely necessary, to design a discrete voltage amplifier circuit that intentionally uses this narrow region because of its proximity to hard saturation, the increasing risk of significant distortion, and the impact of temperature and part variations on this mode.
There are still some reasons for discrete designs. For example, when the load needs to be driven using voltages, currents, or at dissipations that are incompatible with routinely available opamps. Or for pre-amplifier stages that must more precisely accommodate their input transducer and deliver the best possible noise performance. Or when you need to take advantage of their exponential/logarithmic behavior (VU meters or log-amps, for example.)
If you are designing a discrete voltage amplifier then it's better to keep the BC junction reverse-biased for the entire signal and to do so for all intended operating temperatures and for variability in parts that will be used in the circuit. If you are designing for a switching application, then hard saturation is the way to go. Soft saturation is generally to be avoided. Temperature and part variations alone will cause no end of trouble in this case, if nothing else is said about it.