The definition is rather interesting, because its based on the models of the transistor, not the transistor itself. Obviously a real transistor has smooth responses to all inputs, with no magic "points" where its behavior suddenly changes. However, that response is complicated. We observe that, once Vbe goes above a certain voltage (0.7V for many transistors), we can use drastically simpler equations to model the behavior and still achieve very accurate results. Thus, datasheets typically give a "forward bias" Vbe which is the bias needed to get the transistor into that mode.
Under the hood, transistors are closely related to diodes, so we can use the diode equations to understand where some of these behaviors come from. The Shockley equation for a diode is \$ I=I_s(e^\frac{V_D}{nV{T}}-1) \$ As we can see, we have an exponential minus a constant, so for small or negative \$V_D\$, the exponential will be very close to 1, so the subtraction will result in almost zero current across the diode. However, at larger \$V_D\$, the subtraction becomes almost unimportant, and we see an exponential rise in current with voltage. Where does this break occur? It's based on \$V_T\$ which is the thermal voltage (a constant) and \$n\$, which is an "ideality" factor (typically between 1 and 2). Needless to say, the mere fact that there is such an "ideality" fudge factor in the equation shows why this concept of a forward bias is more empirical than anything else.
So really, if you think about it, the concept of forward bias divides the behavior of the transistor into three regions, including some \$\epsilon\$ bound around \$V_{th}\$ to account for the murky transition:
- Forward biased (\$V_{BE} > V_{th} + \epsilon\$): equations may be simplified by ignoring most of the low voltage effects.
- Reverse biased (\$V_{BE} < V_{th} - \epsilon\$): equations may assume the transistor is "off," where the (exponential-1) is very close to 0.
- Transition (\$V_{BE} \approx V_{th}\$): neither simplified equation will really accurately model the behaviors in this region. If you're really really close to \$V_{th}\$, you may see strange artifacts which are only explained by a more complete transistor model which is not dependent on \$V_{th}\$ as a fundamental parameter.
Note, there is an effect called avalanche breakdown, where the diode ceases to prevent current flow (due to quantum effects) if you put a high enough negative voltage on it. Its behavior is also well modeled by an exponential, so you have another fuzzy region where you need a more complicated model, after which the equations simplify again. This suggests five total regions (from most negative \$V_{BE}\$ to most positive): breakdown, transition 1, reverse biased, transition 2, forward biased. Fortunately for most cases, you can ignore the transitions because they are small enough, but if you want to understand precisely what is happening, it helps to remember that they are there.
