Input impedance for a darlington pair
It seems then that the equation in the video and in Floyd’s book differ by what assumptions they take. 
The input impedance presented in Floyd uses the special case where beta of both transistors are the same. Also in the book they ignore re1. 

Two other books I have shown an equation of 
Zin=beta1*beta2*Re where Re=RE||RL. There they make the assumption that re1 and re2 terms are small compared to RE. 

Shown below is a capture of one such equations derived from one of these books

Source: electronic devices and circuit theory boylestad 



So the equation you use really depends on
what assumptions you want to take and how close you want to to be to represent each piece of the model. If that makes sense.