EDITED: I've found a reference with a precise explanation for the use of the word "common".
Source: A Practical Introduction to Electronic Circuits, Martin Hartley JONES, 3ed., 1995.
"As the 'name common collector" suggests, the transistor collector is connected straight to the supply line which, as far as the signals are concerned, is the same thing as the earth (common) rail, because power supply outputs are always designed to present a very low impedance to signals. The output load resistor, RL, is in the emitter circuit, whilst the input signal is fed in between base and earth in the usual way."
Furthermore, HOROWITZ and HILL(The Art of Electronics, 2nd Edition) don't even call this kind of circuit as "common collector". They call it just "emitter follower".
"It is called that because the output terminal is the emitter, which
follows the input (the base), less one diode drop:
VE ≈ VB - 0.6 volt
The output is a replica of the input, but 0.6 to 0.7 volt less
positive. For this circuit, Vin must stay at +0.6V or more, or else
the output will sit at ground. By returning the emitter resistor to a negative supply voltage, you can permit negative voltage swings as well. Note that there is no collector resistor in an emitter follower:
At first glance this circuit may appear useless, until you realize
that the input impedance is much larger than the output impedance
In other words, an emitter follower has no current gain, even
though it has no voltage gain. It has power gain. Voltage gain isn't
Both books (JONES, and also HOROWITZ & HILL) also refer to a "common base" circuit, and in those "common base" circuits we see the base of a BJT tied to one of the power-supply rails (positive or negative), the input signal is fed to the emitter, and the output is taken from the collector. This configuration is used at high frequencies.
My conclusion is that allaboutcircuits.com made a mistake when they said that "It is called the common-collector configuration because (ignoring the power supply battery) both the signal source and the load share the collector lead as a common connection point" , because the word "common" is exactly due to the power supply +vcc connection. The @Nasha's and @Dirac16's answers to this question also speaks about the connection to the power supply in order to define "common".
So, it seems that the word "common" is used because the so-called "common" terminal is connected to a fixed voltage (+Vcc or GND) with a very low impedance to signals, and not because it is shared by the input and output.