Change in Consumer Surplus with Two Price Changes: Is it Path Dependent?
Your reasoning is correct (i.e. the book is wrong). First, let's follow the book's logic a little more carefully step by step, beginning with the case where p1 changes first:


a fall in p1 leads to a gain of A in market 1 and of H in market 2 (because D2 shifts).
a subsequent fall in p2 leads to a gain of B + C in market 2 (we are now using the new demand curve in market 2); moreover, it causes D1 to shift so the consumer gains G + D.
thus, the total gain is A + H +G + D + B + C.


If p2 changes first:


a fall in p2 leads to a gain of B in market 2 and of G in market 1 (because D1 shifts).
a subsequent fall in p1 leads to a gain of A + D in market 1 (we are now using the new demand curve in market 1); moreover, it causes D2 to shift so the consumer gains H + C.
thus, the total gain is A + H +G + D + B + C.


We see that the order doesn't matter.

Another quick way to verify that the order doesn't matter is to note that these demand curves are coming from a consumer who chooses the optimal demand vector for given prices. If the prices are the same then the solution to the maximisation problem will also be the same.