Understanding common terminal in common collector/emitter amplifier
A BJT transistor has three terminals, namely Base(B), Emitter(E), and Collector(C). But we need actually four terminals to connect the transistor to the rest of the circuit. So we need two terminals for the input and two terminals for the output. In a common collector configuration, which is also known as emitter follower, the collector terminal is shared between the input and output signals, as shown in your schematic. If you AC ground the collector, i.e. short out the DC battery source, you would end up with the collector standing as the common terminal for both input and output signals. 

The purpose of common collector is to buffer an AC input signal, not DC signal. Also note that a BJT is mostly used for amplification. So the name has something do with AC signals. A DC input signal, however, is just used for biasing the transistor. So it has no other purposes. Therefore, the name common collector comes after we AC ground all signals present in the circuit. If you AC ground the input signal, which is an AC signal, such as voice, plus a DC bias voltage, you would end up with the AC signal itself, the DC bias part would be shorted. The AC analysis of the battery like the DC bias voltage would be shorted. Therefore at the end you'll end up with the collector as the common terminal between the input and output signals. 

Here's a schematic that might better show what's going on: