Could the destinction between early morning coughing and rest of day coughing be clarified: how would the patients be asked to differentiate between immediately upon rising vs. rest of day?

The typical early morning coughing starts as soon as (within a couple of minutes of) them waking up or getting up and is almost always productive. Anything else is rest of the day. Patients with the typical early morning cough can usually easily distinguish the two types.