I really like your method of determining common collector as "transistor lead not in the signal path" because it is so straight forward. However, I have always understood it to be the lead which is common to both input and output sides of the transistor circuit. For a common collector amplifier, this would be the collector since it's connected to V+ (power) which is common to both input and output of the circuit. Both methods lead to the same result and I suspect the mapping between the two methods is that if a lead is common, it can't be in the signal path since it is providing power or ground to both the input and output sides.
MOST EXCELLENT! I'd use a unity gain gain stage mainly for buffering and or sometimes a 180 degree phase inversion of the signal so I can blend, or combine or remix two similar very signals derived from one original input signal where one signal is 180 degrees out from the other. I had one that used two BJTs and a voltage divider off of a cathode follower triode for tube amplification phase inversion of the signal to remix the same signal back in. I did another tube amp phase inverting Op Amp (not a phase splitter). The op amp circuit required a switch that dumped stored DC from coupling caps to ground to protect the op amp when switching it in. It was two tube preamps from two pretty infamous devices to put in front of a guitar amp.
I always heard transistors configured like this called an Emitter Follower. It makes sense because the emitter voltage follows the base voltage via the base-emitter diode in the transistor but with the current gain of the transitor giving smaller output impedance.
