I think that a momentary closure of I1 should only bypass (or skip) the Q2 Output ON periods.
I1 ON should also cause the timers that control Q2 to be skipped, so that there are not waiting periods with no purpose.
You got it, this is basically how I envisioned it might work. Otherwise the timers carry on, hence the Q2 output would resume (until the timer had finished). This what I saw happening yesterday.