In a common-emitter amplifier, under which conditions are we supposed to use one equation over the other?
Carl - I think, the answer is simple: You have shown the correct equation and it depends on the particular values if you can neglect some parameters or not.

The decision if you are allowed to neglect some parameters depends (a) on the overall accuracy requirements and (b) on the tolerances of all parameters involved. For example, it makes no sense to consider in the denominaor a resistor which would improve the accuracy by 5% when - at the same time - the resistors and/or the beta-value have tolerances in the order of 5% or even 10%.

Examples:

For pure voltage input (Rth=0) the denominator within the brackets is (rpi +RE). Furthermore, for RE>>rpi the expression in brackets is (1+beta*RE/rpi). In this case, it is very convenient to remember that the ratio beta/rpi is identical to the transconductance gm=beta/rpi. In this case, we are not dependent anymore on the uncertain parameters beta and rpi because their ratio gm can be found simply by gm=Ic/Vt. Now the simplified formula as given by you (in yellow) applies. (Perhaps it is more accurat to say but "may be used" instead of "applies",