I'm sure you know that you should be very careful using a trolley jack on carpet!
The trolley should be free to move so that as the point of lift moves backwards (as the arm swings up) the trolley tends to want to move forward to stay directly under the weight. If the car cannot move and the jack is snagged on the carpet it could be dangerous!
Bottle jacks and scissor jacks just raise directly upwards, a trolley jack swings backwards on its arm as it is raised.
The trolley jack does move slightly when the car is being raised on the carpet, the arm swings as you say, and the block remains central (Block routered into the jacks lifting point)
The handbrake is off so the car also moves slightly as it’s being raised. Once it’s fully raised then the chocks go behind the rear wheels inc axel stands in place.