You might be interested that this research into symptoms also doesn’t mention a “type” of cough covid.joinzoe.com/post/covid-clusters
The six clusters are as follows:
1 (‘flu-like’ with no fever): Headache, loss of smell, muscle pains, cough, sore throat, chest pain, no fever.
2 (‘flu-like’ with fever): Headache, loss of smell, cough, sore throat, hoarseness, fever, loss of appetite.
3 (gastrointestinal): Headache, loss of smell, loss of appetite, diarrhea, sore throat, chest pain, no cough.
4 (severe level one, fatigue): Headache, loss of smell, cough, fever, hoarseness, chest pain, fatigue.
5 (severe level two, confusion): Headache, loss of smell, loss of appetite, cough, fever, hoarseness, sore throat, chest pain, fatigue, confusion, muscle pain.
6 (severe level three, abdominal and respiratory): Headache, loss of smell, loss of appetite, cough, fever, hoarseness, sore throat, chest pain, fatigue, confusion, muscle pain, shortness of breath, diarrhea, abdominal pain.