dddd

You mentioned that bit of movement of the shifter and derailer you can make while engaged in one gear. That's called overshift, but it should always spring back to the same position after you make that slight movement of the lever as if trying to downshift to a bigger cog.
If the derailer can hang at more than one position for each indexed position of the lever, then there is too much friction in the shifter or the cabling. The overshift is meant to be only momentary during downshifts, and if the derailer doesn't freely spring back to the exact indexed position then this will happen when the frame sees the highest pedaling force, as when accelerating from a stop as you described. That can cause at least a partial, momentary ghost-shift that sounds just like what you've described.
I would first tug on the exposed cable wire along the downtube to check for cable friction and smooth derailer movement and return. If that seems smooth, I would remove the lever and blast it out with aerosol (foaming) Tri-Flow (look for a tiny spring-end opening on the lever's back side), then re-mount and check shifter movement before and as the mounting bolt is secured. The overshift movement should now easily be overcome by the cable's tension alone so that the derailer always returns to a precise location for each indexed position, and frame flex can't tug the cable enough to make the lever move.