Yep you fuckk up the distributor from the looks of your video. Hopefully not too bad. Why would you go in there prying about with a screwdriver while the distributor is bolted in place. You need a good rapping on the knuckles with a 1/2" breaker bar & then your nose rubbed in the valve cover oil fill hole whilst repeating "I shall refer to the FSM in the future".
No the dristributor shaft & rotor should not have right or left movement like yours if properly seated. Unbolt the distributor from the front engine cover & inspect for any damage by refering to the FSM specs. I suspect you lifted the distributor shaft off of the oil pump/distributor spindle & its not seating properly now. In any case a thorough off engine inspection of the distributor should be done.
JohnH: Why is the first paragraph of your response so littered with spelling and grammatical errors that it's mostly incomprehensible...but the second is much more readable? Looks like it was written by two different people. Just curious.