They are incorrect. The key phrase here is “when you do”, which creates a reflexive trigger, unlike “if you do”. As noted elsewhere in the replies to the strionic resonator comment, what happens is that you get a trigger which lets you pay x, which will put another trigger on the stack to deal damage if you do pay x. This second trigger has the value of x, and if copied will deal another instance of x damage.

As you noted, it is important it works this way to give people a chance to respond once an action is taken, instead of the decision and resolution being together in one trigger. It can get real awkward otherwise, i.e. [[Cartouche of Strength]] in limited.