Congestion can be anything from stuffed up sinuses to "chest congestion" (having a lot of mucous in the airways).
A "dry cough" is one that tends to "hacky". Whether or not it is a hacking cough, a dry cough doesn't produce anything.
A "chesty" cough may more often be called a "productive cough" . There may be a "loose, rumbling" in the chest and a sense of "clearing out" after coughing. Not all productive coughs include the "loose rumbling" upon breathing, but usually when someone refers to a "chesty" cough that's what he means.