I think this is a very wrong way to look at the configurations. The common node is common to both input and output.
Base is always part of input, collector is always part of output, but:
In a common emitter configuration, the input is applied across base to emitter, and the output is taken across collector and emitter.
In the common base configuration, the input is applied emitter to base, and the output is taken across base to collector.
In the common collector configuration, the input signal is applied across base to collector, and the output is taken across collector to emitter.
The common terminal is just the one that essentially holds a constant voltage (an AC common) while the other two swing with the input and output signals.