It was an attempt at an over simplified explanation. (and I spelled "fouling" wrong) Your details are correct, but there is still no "piston" in an AR unless it is a "piston" AR. "Like" a piston is not a piston, ALL self loading firearms have parts that are "like" a piston.
The piston and the bolt are the same piece in the AR - what is a piston? the part where gas actuated the system by exerting force on it. Voila, that is the second half of the bolt in an AR - that's why it even comes with gas ring to seal up the gas.
Think of the Bolt carrier and the bolt form an expandable gas chamber.


















































