Active Band Pass Filter Design
The trick is to know which terms in your derived equation are significant and which are not. The original designer's equations are simplifications that make the assumption that the various cutoff frequencies are "far enough" from each other that they don't significantly affect each other. Your "full" derivation includes terms for those interactions. There's really no rigorous mathematical way to get from the latter to the former; you mainly need to put real values in for the components and then decide whether each term is large enough to matter.

Bottom line: Use the designer's equations to help you pick component values, then use your equations to see what the actual response is going to be. Make small adjustments to the values as needed.