Why did my textbook use signal energy as a measurement of approximation error, instead of something else?
Using all the area under a data bit signal (integrating) and its energy gives the maximum signal when compared to the noise error. Less performing methods use peak signal and then peak noise may apply but gives worse SNR and higher BER.  Good demodulators may be harder than center peak sample but integrating the entire signal energy over the symbol gives higher SNR and lower BER.

When you get into ADC specs they use about 6 different ways to express error  for quantifying ADC dynamic performance.   

SINAD (signal-to-noise-and-distortion ratio),
 ENOB (effective number of bits),
 SNR (signal-to-noise ratio),
 THD (total harmonic distortion),
 THD + N (total harmonic distortion plus noise),
 SFDR (spurious free dynamic range).