Which is actually an active component?

There is a clear definition: Passive elements have no function of gain, or control over voltage or current: their controlling function is linear -> I/V = R in the case of a resistor. There are exactly four kinds of passive elements: Resistors, Capacitors, Inductors and Memristors. All other components are active. Source http://de.wikipedia.org/wiki/Elektrisches_Bauelement

Active Elements have a function of gain or control, meaning the connection of the controlling parameters nonlinear. Diodes control current, transistors amplify current, etc.

The reason for the distinction is mathematical: You can use certain mathematical approaches to solve the equations of a device that contains only passive elements, while the same approaches would not work with active elements. If you have active elements, you may have to first approximate a passive network at the working conditions before you calculate.

This does not mean you cannot build advanced devices out of passive Elements. Analog filters are often made from passive elements and can be quite complicated.

Inverting the collector current equation:

$$i_C = I_Se^{\frac{v_{BE}}{V_T}}$$

yields:

$$v_{BE} = V_T\ln{\frac{i_C}{I_S}}$$

For example, let

$$V_T = 25mV$$

$$I_S = 1 fA$$

$$I_C = 1mA$$

With these values, find that

$$V_{BE} = 0.691V$$

Now, double the collector current and find that

$$V_{BE} = 0.708V$$

Increasing the collector current 100% only increased the base-emitter voltage 2.45%

So, while it is not true that the base-emitter voltage is constant, it is not a bad approximation to consider it constant over a relatively wide range of collector current.