Hmm, if the experiment is set to save “Final state” rather than a click it seems to me that we wouldn’t expect a reaction time to be saved. In that case the online experiment is doing the right thing, and the local experiment is doing the wrong thing! The final state suggests to me “What was the mouse state at the end of the routine?” so it says whether the buttons at that point were up/down and what was the location.
I agree with David that your solution is to set it to save the click first/last/all if you want an RT because RT is necessarily related to clicks.