It seems to me that a transistor is merely controlling a greater voltage or current with its own smaller voltage and current. Am I missing something? I can't expect to take a single little 5V 1A power supply and through a transistor turn it into 12V or 5A, right?
You are correct.
It sounds to me you are stuck on the definition of amplification as "literally" increasing the power of the signal. It does not. For all the context's I've seen/heard of amplification there are usually two "givens or assumptions"
- There is some power source supplying the energy to create the amplified signal
- There is a continuous capability. By that I mean your relay example is more properly termed a "switching" action as opposed to an amplification action. Transistors and hydraulic spool valves and audio amplifiers and..... all can provide various levels of amplification. A relay cannot.